3.5.75 \(\int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx\) [475]

Optimal. Leaf size=284 \[ -\frac {b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt {c+d x}}{8 a^4 (a+b x)}+\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^5 \sqrt {c}}-\frac {\sqrt {b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^5} \]

[Out]

-1/3*c*(d*x+c)^(3/2)/a/x^3/(b*x+a)-(-3*a*d+8*b*c)*(-a*d+b*c)^(3/2)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1
/2))*b^(1/2)/a^5+1/8*(-5*a^3*d^3+60*a^2*b*c*d^2-120*a*b^2*c^2*d+64*b^3*c^3)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^5
/c^(1/2)-1/8*b*(19*a^2*d^2-52*a*b*c*d+32*b^2*c^2)*(d*x+c)^(1/2)/a^4/(b*x+a)+1/12*c*(-9*a*d+8*b*c)*(d*x+c)^(1/2
)/a^2/x^2/(b*x+a)-1/24*(33*a^2*d^2-82*a*b*c*d+48*b^2*c^2)*(d*x+c)^(1/2)/a^3/x/(b*x+a)

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Rubi [A]
time = 0.28, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 154, 156, 162, 65, 214} \begin {gather*} -\frac {\sqrt {b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^5}+\frac {c \sqrt {c+d x} (8 b c-9 a d)}{12 a^2 x^2 (a+b x)}-\frac {b \sqrt {c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{8 a^4 (a+b x)}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{24 a^3 x (a+b x)}+\frac {\left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^5 \sqrt {c}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]

[Out]

-1/8*(b*(32*b^2*c^2 - 52*a*b*c*d + 19*a^2*d^2)*Sqrt[c + d*x])/(a^4*(a + b*x)) + (c*(8*b*c - 9*a*d)*Sqrt[c + d*
x])/(12*a^2*x^2*(a + b*x)) - ((48*b^2*c^2 - 82*a*b*c*d + 33*a^2*d^2)*Sqrt[c + d*x])/(24*a^3*x*(a + b*x)) - (c*
(c + d*x)^(3/2))/(3*a*x^3*(a + b*x)) + ((64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sq
rt[c + d*x]/Sqrt[c]])/(8*a^5*Sqrt[c]) - (Sqrt[b]*(8*b*c - 3*a*d)*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d
*x])/Sqrt[b*c - a*d]])/a^5

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx &=-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (8 b c-9 a d)+\frac {1}{2} d (5 b c-6 a d) x\right )}{x^3 (a+b x)^2} \, dx}{3 a}\\ &=\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}-\frac {\int \frac {-\frac {1}{4} c \left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right )-\frac {1}{4} d \left (40 b^2 c^2-65 a b c d+24 a^2 d^2\right ) x}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx}{6 a^2}\\ &=\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\int \frac {-\frac {3}{8} c \left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right )-\frac {3}{8} b c d \left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) x}{x (a+b x)^2 \sqrt {c+d x}} \, dx}{6 a^3 c}\\ &=-\frac {b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt {c+d x}}{8 a^4 (a+b x)}+\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\int \frac {-\frac {3}{8} c (b c-a d) \left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right )-\frac {3}{8} b c d (b c-a d) \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{6 a^4 c (b c-a d)}\\ &=-\frac {b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt {c+d x}}{8 a^4 (a+b x)}+\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\left (b (8 b c-3 a d) (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^5}-\frac {\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx}{16 a^5}\\ &=-\frac {b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt {c+d x}}{8 a^4 (a+b x)}+\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\left (b (8 b c-3 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^5 d}-\frac {\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 a^5 d}\\ &=-\frac {b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt {c+d x}}{8 a^4 (a+b x)}+\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^5 \sqrt {c}}-\frac {\sqrt {b} (8 b c-3 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^5}\\ \end {align*}

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Mathematica [A]
time = 0.98, size = 224, normalized size = 0.79 \begin {gather*} \frac {-\frac {a \sqrt {c+d x} \left (96 b^3 c^2 x^3+12 a b^2 c x^2 (4 c-13 d x)+a^3 \left (8 c^2+26 c d x+33 d^2 x^2\right )+a^2 b x \left (-16 c^2-82 c d x+57 d^2 x^2\right )\right )}{x^3 (a+b x)}+24 \sqrt {b} (8 b c-3 a d) (-b c+a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )+\frac {3 \left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{24 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]

[Out]

(-((a*Sqrt[c + d*x]*(96*b^3*c^2*x^3 + 12*a*b^2*c*x^2*(4*c - 13*d*x) + a^3*(8*c^2 + 26*c*d*x + 33*d^2*x^2) + a^
2*b*x*(-16*c^2 - 82*c*d*x + 57*d^2*x^2)))/(x^3*(a + b*x))) + 24*Sqrt[b]*(8*b*c - 3*a*d)*(-(b*c) + a*d)^(3/2)*A
rcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]] + (3*(64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*
d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/Sqrt[c])/(24*a^5)

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Maple [A]
time = 0.10, size = 290, normalized size = 1.02

method	result	size
derivativedivides	\(2 d^{5} \left (-\frac {\left (a d -b c \right )^{2} b \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (3 a d -8 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{5} d^{5}}+\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{4} a^{2} b c \,d^{2}+\frac {3}{2} a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+4 a^{2} b \,c^{2} d^{2}-3 a \,b^{2} c^{3} d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{4} a^{2} b \,c^{3} d^{2}+\frac {3}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {d x +c}}{d^{3} x^{3}}-\frac {\left (5 a^{3} d^{3}-60 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{5} d^{5}}\right )\)	\(290\)
default	\(2 d^{5} \left (-\frac {\left (a d -b c \right )^{2} b \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (3 a d -8 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{5} d^{5}}+\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{4} a^{2} b c \,d^{2}+\frac {3}{2} a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+4 a^{2} b \,c^{2} d^{2}-3 a \,b^{2} c^{3} d \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7}{4} a^{2} b \,c^{3} d^{2}+\frac {3}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {d x +c}}{d^{3} x^{3}}-\frac {\left (5 a^{3} d^{3}-60 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{5} d^{5}}\right )\)	\(290\)
risch	\(-\frac {\sqrt {d x +c}\, \left (33 a^{2} d^{2} x^{2}-108 a b c d \,x^{2}+72 b^{2} c^{2} x^{2}+26 a^{2} c d x -24 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 a^{4} x^{3}}-\frac {d^{3} b \sqrt {d x +c}}{a^{2} \left (b d x +a d \right )}+\frac {2 d^{2} b^{2} \sqrt {d x +c}\, c}{a^{3} \left (b d x +a d \right )}-\frac {d \,b^{3} \sqrt {d x +c}\, c^{2}}{a^{4} \left (b d x +a d \right )}-\frac {3 d^{3} b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {14 d^{2} b^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c}{a^{3} \sqrt {\left (a d -b c \right ) b}}-\frac {19 d \,b^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{2}}{a^{4} \sqrt {\left (a d -b c \right ) b}}+\frac {8 b^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{3}}{a^{5} \sqrt {\left (a d -b c \right ) b}}-\frac {5 d^{3} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 a^{2} \sqrt {c}}+\frac {15 d^{2} \sqrt {c}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b}{2 a^{3}}-\frac {15 d \,c^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b^{2}}{a^{4}}+\frac {8 c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b^{3}}{a^{5}}\)	\(431\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/x^4/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2*d^5*(-(a*d-b*c)^2*b/a^5/d^5*(1/2*a*d*(d*x+c)^(1/2)/(b*(d*x+c)+a*d-b*c)+1/2*(3*a*d-8*b*c)/((a*d-b*c)*b)^(1/2)
*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))+1/a^5/d^5*(-((11/16*a^3*d^3-9/4*a^2*b*c*d^2+3/2*a*b^2*c^2*d)*(d*
x+c)^(5/2)+(-5/6*a^3*c*d^3+4*a^2*b*c^2*d^2-3*a*b^2*c^3*d)*(d*x+c)^(3/2)+(-7/4*a^2*b*c^3*d^2+3/2*a*b^2*c^4*d+5/
16*a^3*c^2*d^3)*(d*x+c)^(1/2))/d^3/x^3-1/16*(5*a^3*d^3-60*a^2*b*c*d^2+120*a*b^2*c^2*d-64*b^3*c^3)/c^(1/2)*arct
anh((d*x+c)^(1/2)/c^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

________________________________________________________________________________________

Fricas [A]
time = 1.29, size = 1482, normalized size = 5.22 \begin {gather*} \left [\frac {24 \, {\left ({\left (8 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{4} + {\left (8 \, a b^{2} c^{3} - 11 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 3 \, {\left ({\left (64 \, b^{4} c^{3} - 120 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4} + {\left (64 \, a b^{3} c^{3} - 120 \, a^{2} b^{2} c^{2} d + 60 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a^{4} c^{3} + 3 \, {\left (32 \, a b^{3} c^{3} - 52 \, a^{2} b^{2} c^{2} d + 19 \, a^{3} b c d^{2}\right )} x^{3} + {\left (48 \, a^{2} b^{2} c^{3} - 82 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \, {\left (8 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}, \frac {48 \, {\left ({\left (8 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{4} + {\left (8 \, a b^{2} c^{3} - 11 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - 3 \, {\left ({\left (64 \, b^{4} c^{3} - 120 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4} + {\left (64 \, a b^{3} c^{3} - 120 \, a^{2} b^{2} c^{2} d + 60 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a^{4} c^{3} + 3 \, {\left (32 \, a b^{3} c^{3} - 52 \, a^{2} b^{2} c^{2} d + 19 \, a^{3} b c d^{2}\right )} x^{3} + {\left (48 \, a^{2} b^{2} c^{3} - 82 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \, {\left (8 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}, -\frac {3 \, {\left ({\left (64 \, b^{4} c^{3} - 120 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4} + {\left (64 \, a b^{3} c^{3} - 120 \, a^{2} b^{2} c^{2} d + 60 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - 12 \, {\left ({\left (8 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{4} + {\left (8 \, a b^{2} c^{3} - 11 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + {\left (8 \, a^{4} c^{3} + 3 \, {\left (32 \, a b^{3} c^{3} - 52 \, a^{2} b^{2} c^{2} d + 19 \, a^{3} b c d^{2}\right )} x^{3} + {\left (48 \, a^{2} b^{2} c^{3} - 82 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \, {\left (8 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}, \frac {24 \, {\left ({\left (8 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{4} + {\left (8 \, a b^{2} c^{3} - 11 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - 3 \, {\left ({\left (64 \, b^{4} c^{3} - 120 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4} + {\left (64 \, a b^{3} c^{3} - 120 \, a^{2} b^{2} c^{2} d + 60 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - {\left (8 \, a^{4} c^{3} + 3 \, {\left (32 \, a b^{3} c^{3} - 52 \, a^{2} b^{2} c^{2} d + 19 \, a^{3} b c d^{2}\right )} x^{3} + {\left (48 \, a^{2} b^{2} c^{3} - 82 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \, {\left (8 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

[1/48*(24*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3
)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 3*((64*b^4*
c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d
^2 - 5*a^4*d^3)*x^3)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(8*a^4*c^3 + 3*(32*a*b^3*c^3 - 5
2*a^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3
- 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), 1/48*(48*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c
*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)
*sqrt(d*x + c)/(b*d*x + b*c)) - 3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a
*b^3*c^3 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2
*c)/x) - 2*(8*a^4*c^3 + 3*(32*a*b^3*c^3 - 52*a^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*
c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), -1/24*
(3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*b^2*c^2*d +
60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - 12*((8*b^3*c^3 - 11*a*b^2*c^2*d +
3*a^2*b*c*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c
- a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + (8*a^4*c^3 + 3*(32*a*b^3*c^3 - 52*a^2*b^2*c^2*d + 1
9*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3 - 13*a^4*c^2*d)*x)*
sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), 1/24*(24*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)*x^4 + (8*a*b^
2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d
*x + b*c)) - 3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*
b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - (8*a^4*c^3 + 3*(32*a*
b^3*c^3 - 52*a^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(8
*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2480 vs. \(2 (270) = 540\).
time = 216.01, size = 2480, normalized size = 8.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/2)/x**4/(b*x+a)**2,x)

[Out]

2*b**4*c**3*d*sqrt(c + d*x)/(2*a**6*d**2 - 2*a**5*b*c*d + 2*a**5*b*d**2*x - 2*a**4*b**2*c*d*x) - 6*b**3*c**2*d
**2*sqrt(c + d*x)/(2*a**5*d**2 - 2*a**4*b*c*d + 2*a**4*b*d**2*x - 2*a**3*b**2*c*d*x) + 6*b**2*c*d**3*sqrt(c +
d*x)/(2*a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x - 2*a**2*b**2*c*d*x) + 20*b*c**4*d**2*sqrt(c + d*x)/(-8*a**
3*c**4 - 16*a**3*c**3*d*x + 8*a**3*c**2*(c + d*x)**2) - 12*b*c**3*d**2*(c + d*x)**(3/2)/(-8*a**3*c**4 - 16*a**
3*c**3*d*x + 8*a**3*c**2*(c + d*x)**2) - 2*b*d**4*sqrt(c + d*x)/(2*a**3*d**2 - 2*a**2*b*c*d + 2*a**2*b*d**2*x
- 2*a*b**2*c*d*x) - 66*c**5*d**3*sqrt(c + d*x)/(96*a**2*c**6 + 144*a**2*c**5*d*x - 144*a**2*c**4*(c + d*x)**2
+ 48*a**2*c**3*(c + d*x)**3) + 80*c**4*d**3*(c + d*x)**(3/2)/(96*a**2*c**6 + 144*a**2*c**5*d*x - 144*a**2*c**4
*(c + d*x)**2 + 48*a**2*c**3*(c + d*x)**3) - 30*c**3*d**3*(c + d*x)**(5/2)/(96*a**2*c**6 + 144*a**2*c**5*d*x -
144*a**2*c**4*(c + d*x)**2 + 48*a**2*c**3*(c + d*x)**3) - 30*c**3*d**3*sqrt(c + d*x)/(-8*a**2*c**4 - 16*a**2*
c**3*d*x + 8*a**2*c**2*(c + d*x)**2) + 18*c**2*d**3*(c + d*x)**(3/2)/(-8*a**2*c**4 - 16*a**2*c**3*d*x + 8*a**2
*c**2*(c + d*x)**2) + b*d**4*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*
d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a) - b*d**4*sqrt(-1/
(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c
**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a) - 3*b**2*c*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d
**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)
) + sqrt(c + d*x))/(2*a**2) + 3*b**2*c*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**
3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a**2)
- 5*c**3*d**3*sqrt(c**(-7))*log(-c**4*sqrt(c**(-7)) + sqrt(c + d*x))/(16*a**2) + 5*c**3*d**3*sqrt(c**(-7))*log
(c**4*sqrt(c**(-7)) + sqrt(c + d*x))/(16*a**2) + 9*c**2*d**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(c +
d*x))/(8*a**2) - 9*c**2*d**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(c + d*x))/(8*a**2) - 3*c*d**3*sqrt(c*
*(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**2) + 3*c*d**3*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqr
t(c + d*x))/(2*a**2) - 2*d**3*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**2*sqrt(a*d/b - c)) + 2*d**3*atan(sqrt(c
+ d*x)/sqrt(-c))/(a**2*sqrt(-c)) - 3*d**2*sqrt(c + d*x)/(a**2*x) + 3*b**3*c**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)
)*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a
*d - b*c)**3)) + sqrt(c + d*x))/(2*a**3) - 3*b**3*c**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/
(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c +
d*x))/(2*a**3) - 3*b*c**3*d**2*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(c + d*x))/(4*a**3) + 3*b*c**3*d**
2*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(c + d*x))/(4*a**3) + 3*b*c**2*d**2*sqrt(c**(-3))*log(-c**2*sqrt(
c**(-3)) + sqrt(c + d*x))/a**3 - 3*b*c**2*d**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x))/a**3 + 12
*b*c*d**2*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**3*sqrt(a*d/b - c)) - 12*b*c*d**2*atan(sqrt(c + d*x)/sqrt(-c)
)/(a**3*sqrt(-c)) + 6*b*c*d*sqrt(c + d*x)/(a**3*x) - b**4*c**3*d*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sq
rt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sq
rt(c + d*x))/(2*a**4) + b**4*c**3*d*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*
a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*a**4) - 3*b**2
*c**3*d*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**4) + 3*b**2*c**3*d*sqrt(c**(-3))*log(c**2
*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a**4) - 18*b**2*c**2*d*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**4*sqrt(a*d/b
- c)) + 18*b**2*c**2*d*atan(sqrt(c + d*x)/sqrt(-c))/(a**4*sqrt(-c)) - 3*b**2*c**2*sqrt(c + d*x)/(a**4*x) + 8*
b**3*c**3*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**5*sqrt(a*d/b - c)) - 8*b**3*c**3*atan(sqrt(c + d*x)/sqrt(-c)
)/(a**5*sqrt(-c))

________________________________________________________________________________________

Giac [A]
time = 0.49, size = 370, normalized size = 1.30 \begin {gather*} \frac {{\left (8 \, b^{4} c^{3} - 19 \, a b^{3} c^{2} d + 14 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{5}} - \frac {{\left (64 \, b^{3} c^{3} - 120 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{8 \, a^{5} \sqrt {-c}} - \frac {\sqrt {d x + c} b^{3} c^{2} d - 2 \, \sqrt {d x + c} a b^{2} c d^{2} + \sqrt {d x + c} a^{2} b d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a^{4}} - \frac {72 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d - 144 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d + 72 \, \sqrt {d x + c} b^{2} c^{4} d - 108 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c d^{2} + 192 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 84 \, \sqrt {d x + c} a b c^{3} d^{2} + 33 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} c d^{3} + 15 \, \sqrt {d x + c} a^{2} c^{2} d^{3}}{24 \, a^{4} d^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

(8*b^4*c^3 - 19*a*b^3*c^2*d + 14*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sq
rt(-b^2*c + a*b*d)*a^5) - 1/8*(64*b^3*c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*arctan(sqrt(d*x + c)
/sqrt(-c))/(a^5*sqrt(-c)) - (sqrt(d*x + c)*b^3*c^2*d - 2*sqrt(d*x + c)*a*b^2*c*d^2 + sqrt(d*x + c)*a^2*b*d^3)/
(((d*x + c)*b - b*c + a*d)*a^4) - 1/24*(72*(d*x + c)^(5/2)*b^2*c^2*d - 144*(d*x + c)^(3/2)*b^2*c^3*d + 72*sqrt
(d*x + c)*b^2*c^4*d - 108*(d*x + c)^(5/2)*a*b*c*d^2 + 192*(d*x + c)^(3/2)*a*b*c^2*d^2 - 84*sqrt(d*x + c)*a*b*c
^3*d^2 + 33*(d*x + c)^(5/2)*a^2*d^3 - 40*(d*x + c)^(3/2)*a^2*c*d^3 + 15*sqrt(d*x + c)*a^2*c^2*d^3)/(a^4*d^3*x^
3)

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Mupad [B]
time = 1.11, size = 2151, normalized size = 7.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2)/(x^4*(a + b*x)^2),x)

[Out]

(((c + d*x)^(3/2)*(40*a^3*c*d^4 - 288*b^3*c^4*d + 564*a*b^2*c^3*d^2 - 319*a^2*b*c^2*d^3))/(24*a^4) - ((c + d*x
)^(5/2)*(33*a^3*d^4 - 288*b^3*c^3*d + 516*a*b^2*c^2*d^2 - 253*a^2*b*c*d^3))/(24*a^4) + ((c + d*x)^(1/2)*(32*b^
3*c^5*d - 5*a^3*c^2*d^4 - 68*a*b^2*c^4*d^2 + 41*a^2*b*c^3*d^3))/(8*a^4) - (b*d*(c + d*x)^(7/2)*(19*a^2*d^2 + 3
2*b^2*c^2 - 52*a*b*c*d))/(8*a^4))/(b*(c + d*x)^4 - (4*b*c^3 - 3*a*c^2*d)*(c + d*x) + (6*b*c^2 - 3*a*c*d)*(c +
d*x)^2 + (a*d - 4*b*c)*(c + d*x)^3 + b*c^4 - a*c^3*d) + (atan((((((c + d*x)^(1/2)*(601*a^6*b^3*d^8 + 8192*b^9*
c^6*d^2 - 34816*a*b^8*c^5*d^3 - 5976*a^5*b^4*c*d^7 + 59520*a^2*b^7*c^4*d^4 - 52160*a^3*b^6*c^3*d^5 + 24640*a^4
*b^5*c^2*d^6))/(32*a^8) - ((((5*a^13*b^2*d^6)/2 - (41*a^12*b^3*c*d^5)/2 - 16*a^10*b^5*c^3*d^3 + 34*a^11*b^4*c^
2*d^4)/a^12 - ((256*a^11*b^2*d^3 - 512*a^10*b^3*c*d^2)*(c + d*x)^(1/2)*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2
*d - 60*a^2*b*c*d^2))/(512*a^13*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*a^5
*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2)*1i)/(16*a^5*c^(1/2)) + ((((c + d*x)^(1/
2)*(601*a^6*b^3*d^8 + 8192*b^9*c^6*d^2 - 34816*a*b^8*c^5*d^3 - 5976*a^5*b^4*c*d^7 + 59520*a^2*b^7*c^4*d^4 - 52
160*a^3*b^6*c^3*d^5 + 24640*a^4*b^5*c^2*d^6))/(32*a^8) + ((((5*a^13*b^2*d^6)/2 - (41*a^12*b^3*c*d^5)/2 - 16*a^
10*b^5*c^3*d^3 + 34*a^11*b^4*c^2*d^4)/a^12 + ((256*a^11*b^2*d^3 - 512*a^10*b^3*c*d^2)*(c + d*x)^(1/2)*(5*a^3*d
^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(512*a^13*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c
^2*d - 60*a^2*b*c*d^2))/(16*a^5*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2)*1i)/(16*
a^5*c^(1/2)))/(((285*a^8*b^3*d^11)/32 + 512*b^11*c^8*d^3 - 3008*a*b^10*c^7*d^4 - (2765*a^7*b^4*c*d^10)/16 + 74
96*a^2*b^9*c^6*d^5 - 10285*a^3*b^8*c^5*d^6 + (33701*a^4*b^7*c^4*d^7)/4 - (8333*a^5*b^6*c^3*d^8)/2 + (38085*a^6
*b^5*c^2*d^9)/32)/a^12 - ((((c + d*x)^(1/2)*(601*a^6*b^3*d^8 + 8192*b^9*c^6*d^2 - 34816*a*b^8*c^5*d^3 - 5976*a
^5*b^4*c*d^7 + 59520*a^2*b^7*c^4*d^4 - 52160*a^3*b^6*c^3*d^5 + 24640*a^4*b^5*c^2*d^6))/(32*a^8) - ((((5*a^13*b
^2*d^6)/2 - (41*a^12*b^3*c*d^5)/2 - 16*a^10*b^5*c^3*d^3 + 34*a^11*b^4*c^2*d^4)/a^12 - ((256*a^11*b^2*d^3 - 512
*a^10*b^3*c*d^2)*(c + d*x)^(1/2)*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(512*a^13*c^(1/2
)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*a^5*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 1
20*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*a^5*c^(1/2)) + ((((c + d*x)^(1/2)*(601*a^6*b^3*d^8 + 8192*b^9*c^6*d^2 -
34816*a*b^8*c^5*d^3 - 5976*a^5*b^4*c*d^7 + 59520*a^2*b^7*c^4*d^4 - 52160*a^3*b^6*c^3*d^5 + 24640*a^4*b^5*c^2*d
^6))/(32*a^8) + ((((5*a^13*b^2*d^6)/2 - (41*a^12*b^3*c*d^5)/2 - 16*a^10*b^5*c^3*d^3 + 34*a^11*b^4*c^2*d^4)/a^1
2 + ((256*a^11*b^2*d^3 - 512*a^10*b^3*c*d^2)*(c + d*x)^(1/2)*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^
2*b*c*d^2))/(512*a^13*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*a^5*c^(1/2)))
*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*a^5*c^(1/2))))*(5*a^3*d^3 - 64*b^3*c^3 + 120
*a*b^2*c^2*d - 60*a^2*b*c*d^2)*1i)/(8*a^5*c^(1/2)) + (atanh((75*b^3*d^9*(c + d*x)^(1/2)*(b^4*c^3 - a^3*b*d^3 +
3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)^(1/2))/(32*((75*a^2*b^3*d^11)/32 + (811*b^5*c^2*d^9)/32 - (41*b^6*c^3*d^8)/(
2*a) + (6*b^7*c^4*d^7)/a^2 - (211*a*b^4*c*d^10)/16)) + (6*b^5*c^2*d^7*(c + d*x)^(1/2)*(b^4*c^3 - a^3*b*d^3 + 3
*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)^(1/2))/((75*a^4*b^3*d^11)/32 + 6*b^7*c^4*d^7 - (41*a*b^6*c^3*d^8)/2 - (211*a^3
*b^4*c*d^10)/16 + (811*a^2*b^5*c^2*d^9)/32) - (17*b^4*c*d^8*(c + d*x)^(1/2)*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c
*d^2 - 3*a*b^3*c^2*d)^(1/2))/(2*((75*a^3*b^3*d^11)/32 - (41*b^6*c^3*d^8)/2 + (811*a*b^5*c^2*d^9)/32 - (211*a^2
*b^4*c*d^10)/16 + (6*b^7*c^4*d^7)/a)))*(3*a*d - 8*b*c)*(-b*(a*d - b*c)^3)^(1/2))/a^5

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