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

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

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Rubi [A]  time = 0.40, 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 = {98, 149, 151, 156, 63, 208} \begin {gather*} -\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 (60 a^2 b c d^2-5 a^3 d^3-120 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 \sqrt {c+d x} (8 b c-9 a d)}{12 a^2 x^2 (a+b x)}-\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 (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]

-(b*(32*b^2*c^2 - 52*a*b*c*d + 19*a^2*d^2)*Sqrt[c + d*x])/(8*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[Sqrt
[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 63

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 98

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 149

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] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

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] && LtQ[m, -1] && IntegerQ[m]

Rule 156

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, 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 ) \operatorname {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 ) \operatorname {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 = 0.68, size = 237, normalized size = 0.83 \begin {gather*} -\frac {24 \sqrt {b} \sqrt {b c-a d} \left (3 a^2 d^2-11 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )+\frac {a \sqrt {c+d x} \left (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 )+12 a b^2 c x^2 (4 c-13 d x)+96 b^3 c^2 x^3\right )}{x^3 (a+b x)}-\frac {3 \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 )}{\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]

-1/24*((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)) - (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*Sqrt[b]*Sqrt[b*c - a*d]*(8*b^2*c^2 - 11*a*b*c*d
+ 3*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/a^5

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IntegrateAlgebraic [A]  time = 1.10, size = 436, normalized size = 1.54 \begin {gather*} \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 {\sqrt {c+d x} \left (15 a^3 c^2 d^3-40 a^3 c d^3 (c+d x)+33 a^3 d^3 (c+d x)^2-123 a^2 b c^3 d^2+319 a^2 b c^2 d^2 (c+d x)-253 a^2 b c d^2 (c+d x)^2+57 a^2 b d^2 (c+d x)^3+204 a b^2 c^4 d-564 a b^2 c^3 d (c+d x)+516 a b^2 c^2 d (c+d x)^2-156 a b^2 c d (c+d x)^3-96 b^3 c^5+288 b^3 c^4 (c+d x)-288 b^3 c^3 (c+d x)^2+96 b^3 c^2 (c+d x)^3\right )}{24 a^4 d^2 x^3 (a d+b (c+d x)-b c)}+\frac {\left (3 a^4 \sqrt {b} d^4-17 a^3 b^{3/2} c d^3+33 a^2 b^{5/2} c^2 d^2-27 a b^{7/2} c^3 d+8 b^{9/2} c^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{a^5 (a d-b c)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(Sqrt[c + d*x]*(-96*b^3*c^5 + 204*a*b^2*c^4*d - 123*a^2*b*c^3*d^2 + 15*a^3*c^2*d^3 + 288*b^3*c^4*(c + d*
x) - 564*a*b^2*c^3*d*(c + d*x) + 319*a^2*b*c^2*d^2*(c + d*x) - 40*a^3*c*d^3*(c + d*x) - 288*b^3*c^3*(c + d*x)^
2 + 516*a*b^2*c^2*d*(c + d*x)^2 - 253*a^2*b*c*d^2*(c + d*x)^2 + 33*a^3*d^3*(c + d*x)^2 + 96*b^3*c^2*(c + d*x)^
3 - 156*a*b^2*c*d*(c + d*x)^3 + 57*a^2*b*d^2*(c + d*x)^3))/(a^4*d^2*x^3*(-(b*c) + a*d + b*(c + d*x))) + ((8*b^
(9/2)*c^4 - 27*a*b^(7/2)*c^3*d + 33*a^2*b^(5/2)*c^2*d^2 - 17*a^3*b^(3/2)*c*d^3 + 3*a^4*Sqrt[b]*d^4)*ArcTan[(Sq
rt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(a^5*(-(b*c) + a*d)^(3/2)) + ((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]])/(8*a^5*Sqrt[c])

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fricas [A]  time = 2.30, size = 1482, normalized size = 5.22

result too large to display

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)]

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giac [A]  time = 1.37, 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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maple [B]  time = 0.02, size = 545, normalized size = 1.92 \begin {gather*} -\frac {3 b \,d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{2}}+\frac {14 b^{2} c \,d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{3}}-\frac {19 b^{3} c^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{4}}+\frac {8 b^{4} c^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{5}}-\frac {\sqrt {d x +c}\, b \,d^{3}}{\left (b d x +a d \right ) a^{2}}+\frac {2 \sqrt {d x +c}\, b^{2} c \,d^{2}}{\left (b d x +a d \right ) a^{3}}-\frac {\sqrt {d x +c}\, b^{3} c^{2} d}{\left (b d x +a d \right ) a^{4}}-\frac {5 d^{3} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 a^{2} \sqrt {c}}+\frac {15 b \sqrt {c}\, d^{2} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 a^{3}}-\frac {15 b^{2} c^{\frac {3}{2}} d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{4}}+\frac {8 b^{3} c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{5}}-\frac {5 \sqrt {d x +c}\, c^{2}}{8 a^{2} x^{3}}+\frac {7 \sqrt {d x +c}\, b \,c^{3}}{2 a^{3} d \,x^{3}}-\frac {3 \sqrt {d x +c}\, b^{2} c^{4}}{a^{4} d^{2} x^{3}}+\frac {5 \left (d x +c \right )^{\frac {3}{2}} c}{3 a^{2} x^{3}}-\frac {8 \left (d x +c \right )^{\frac {3}{2}} b \,c^{2}}{a^{3} d \,x^{3}}+\frac {6 \left (d x +c \right )^{\frac {3}{2}} b^{2} c^{3}}{a^{4} d^{2} x^{3}}-\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{8 a^{2} x^{3}}+\frac {9 \left (d x +c \right )^{\frac {5}{2}} b c}{2 a^{3} d \,x^{3}}-\frac {3 \left (d x +c \right )^{\frac {5}{2}} b^{2} c^{2}}{a^{4} d^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^3*b/a^2*(d*x+c)^(1/2)/(b*d*x+a*d)+2*d^2*b^2/a^3*(d*x+c)^(1/2)/(b*d*x+a*d)*c-d*b^3/a^4*(d*x+c)^(1/2)/(b*d*x+
a*d)*c^2-3*d^3*b/a^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)+14*d^2*b^2/a^3/((a*d-b*c)
*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c-19*d*b^3/a^4/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/
((a*d-b*c)*b)^(1/2)*b)*c^2+8*b^4/a^5/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c^3-11/8/
a^2/x^3*(d*x+c)^(5/2)+9/2/d/a^3/x^3*(d*x+c)^(5/2)*b*c-3/d^2/a^4/x^3*(d*x+c)^(5/2)*b^2*c^2+5/3/a^2/x^3*(d*x+c)^
(3/2)*c-8/d/a^3/x^3*(d*x+c)^(3/2)*b*c^2+6/d^2/a^4/x^3*(d*x+c)^(3/2)*b^2*c^3+7/2/d/a^3/x^3*(d*x+c)^(1/2)*b*c^3-
3/d^2/a^4/x^3*(d*x+c)^(1/2)*b^2*c^4-5/8/a^2/x^3*(d*x+c)^(1/2)*c^2-5/8*d^3/a^2/c^(1/2)*arctanh((d*x+c)^(1/2)/c^
(1/2))+15/2*d^2/a^3*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b-15*d/a^4*c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b
^2+8/a^5*c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b^3

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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 details)Is a*d-b*c positive or negative?

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mupad [B]  time = 1.11, size = 2151, normalized size = 7.57

result too large to display

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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