3.9.1 \(\int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx\) [801]

Optimal. Leaf size=278 \[ -\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}} \]

[Out]

-1/3*c*(d*x+c)^(3/2)/a/x^3/(b*x+a)^(3/2)+5/8*(-a*d+b*c)*(a^2*d^2-14*a*b*c*d+21*b^2*c^2)*arctanh(c^(1/2)*(b*x+a
)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(11/2)/c^(1/2)-7/24*b*(-7*a*d+15*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^4/(b*x+a)^(3
/2)+3/4*c*(-a*d+b*c)*(d*x+c)^(1/2)/a^2/x^2/(b*x+a)^(3/2)-1/8*(-11*a*d+21*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^3/x/(
b*x+a)^(3/2)-1/24*b*(113*a^2*d^2-420*a*b*c*d+315*b^2*c^2)*(d*x+c)^(1/2)/a^5/(b*x+a)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {100, 154, 156, 157, 12, 95, 214} \begin {gather*} -\frac {7 b \sqrt {c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac {\sqrt {c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}+\frac {3 c \sqrt {c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}}-\frac {b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b*(15*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(24*a^4*(a + b*x)^(3/2)) + (3*c*(b*c - a*d)*Sqrt[c + d*x])/(
4*a^2*x^2*(a + b*x)^(3/2)) - ((21*b*c - 11*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(8*a^3*x*(a + b*x)^(3/2)) - (b*(315
*b^2*c^2 - 420*a*b*c*d + 113*a^2*d^2)*Sqrt[c + d*x])/(24*a^5*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(3*a*x^3*(a
+ b*x)^(3/2)) + (5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sq
rt[c + d*x])])/(8*a^(11/2)*Sqrt[c])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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 157

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] && IntegersQ[2*m, 2*n, 2*p]

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)^{5/2}} \, dx &=-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {9}{2} c (b c-a d)+3 d (b c-a d) x\right )}{x^3 (a+b x)^{5/2}} \, dx}{3 a}\\ &=\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\int \frac {-\frac {3}{4} c (21 b c-11 a d) (b c-a d)-\frac {3}{2} d (9 b c-4 a d) (b c-a d) x}{x^2 (a+b x)^{5/2} \sqrt {c+d x}} \, dx}{6 a^2}\\ &=\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {\int \frac {-\frac {15}{8} c (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )-\frac {3}{2} b c d (21 b c-11 a d) (b c-a d) x}{x (a+b x)^{5/2} \sqrt {c+d x}} \, dx}{6 a^3 c}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {\int \frac {-\frac {45}{16} c (b c-a d)^2 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )-\frac {21}{8} b c d (15 b c-7 a d) (b c-a d)^2 x}{x (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{9 a^4 c (b c-a d)}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {2 \int -\frac {45 c (b c-a d)^3 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a^5 c (b c-a d)^2}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^5}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^5}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}}\\ \end {align*}

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Mathematica [A]
time = 10.29, size = 199, normalized size = 0.72 \begin {gather*} \frac {-8 a^{9/2} c (c+d x)^{7/2}+2 a^{7/2} (9 b c-a d) x (c+d x)^{7/2}-\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) x^2 \left (3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{24 a^{11/2} c^2 x^3 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^(9/2)*c*(c + d*x)^(7/2) + 2*a^(7/2)*(9*b*c - a*d)*x*(c + d*x)^(7/2) - (21*b^2*c^2 - 14*a*b*c*d + a^2*d^2
)*x^2*(3*a^(5/2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*(4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2
)*(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))/(24*a^(11/2)*c^2*x^3*(a + b*x)^(
3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1008\) vs. \(2(234)=468\).
time = 0.08, size = 1009, normalized size = 3.63

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} d^{3} x^{5}-225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c \,d^{2} x^{5}+525 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{4} c^{2} d \,x^{5}-315 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{5} c^{3} x^{5}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b \,d^{3} x^{4}-450 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} c \,d^{2} x^{4}+1050 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{2} d \,x^{4}-630 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{4} c^{3} x^{4}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{5} d^{3} x^{3}-225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b c \,d^{2} x^{3}+525 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d \,x^{3}-315 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} x^{3}+226 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} b^{2} d^{2} x^{4}-840 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a \,b^{3} c d \,x^{4}+630 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{4} c^{2} x^{4}+324 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{3} b \,d^{2} x^{3}-1148 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} b^{2} c d \,x^{3}+840 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a \,b^{3} c^{2} x^{3}+66 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{4} d^{2} x^{2}-192 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{3} b c d \,x^{2}+126 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{2} x^{2}+52 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{4} c d x -36 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} x +16 a^{4} c^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{48 a^{5} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\)	\(1009\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(d*x+c)^(1/2)*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b^2*d^3*x^5-225*ln
((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^3*c*d^2*x^5+525*ln((a*d*x+b*c*x+2*(a*c)^(1
/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^4*c^2*d*x^5-315*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2
)+2*a*c)/x)*b^5*c^3*x^5+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*b*d^3*x^4-450*l
n((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b^2*c*d^2*x^4+1050*ln((a*d*x+b*c*x+2*(a*c)^
(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^3*c^2*d*x^4-630*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^
(1/2)+2*a*c)/x)*a*b^4*c^3*x^4+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^5*d^3*x^3-2
25*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*b*c*d^2*x^3+525*ln((a*d*x+b*c*x+2*(a*c)
^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b^2*c^2*d*x^3-315*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))
^(1/2)+2*a*c)/x)*a^2*b^3*c^3*x^3+226*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*b^2*d^2*x^4-840*((d*x+c)*(b*x+a))
^(1/2)*(a*c)^(1/2)*a*b^3*c*d*x^4+630*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^4*c^2*x^4+324*((d*x+c)*(b*x+a))^(1/
2)*(a*c)^(1/2)*a^3*b*d^2*x^3-1148*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*b^2*c*d*x^3+840*((d*x+c)*(b*x+a))^(1
/2)*(a*c)^(1/2)*a*b^3*c^2*x^3+66*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^4*d^2*x^2-192*((d*x+c)*(b*x+a))^(1/2)*(
a*c)^(1/2)*a^3*b*c*d*x^2+126*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*b^2*c^2*x^2+52*((d*x+c)*(b*x+a))^(1/2)*(a
*c)^(1/2)*a^4*c*d*x-36*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*x+16*a^4*c^2*(a*c)^(1/2)*((d*x+c)*(b*x+a)
)^(1/2))/a^5/((d*x+c)*(b*x+a))^(1/2)/x^3/(a*c)^(1/2)/(b*x+a)^(3/2)

________________________________________________________________________________________

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)^(5/2),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

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Fricas [A]
time = 4.92, size = 848, normalized size = 3.05 \begin {gather*} \left [-\frac {15 \, {\left ({\left (21 \, b^{5} c^{3} - 35 \, a b^{4} c^{2} d + 15 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (21 \, a b^{4} c^{3} - 35 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (21 \, a^{2} b^{3} c^{3} - 35 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{5} c^{3} + {\left (315 \, a b^{4} c^{3} - 420 \, a^{2} b^{3} c^{2} d + 113 \, a^{3} b^{2} c d^{2}\right )} x^{4} + 2 \, {\left (210 \, a^{2} b^{3} c^{3} - 287 \, a^{3} b^{2} c^{2} d + 81 \, a^{4} b c d^{2}\right )} x^{3} + 3 \, {\left (21 \, a^{3} b^{2} c^{3} - 32 \, a^{4} b c^{2} d + 11 \, a^{5} c d^{2}\right )} x^{2} - 2 \, {\left (9 \, a^{4} b c^{3} - 13 \, a^{5} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a^{6} b^{2} c x^{5} + 2 \, a^{7} b c x^{4} + a^{8} c x^{3}\right )}}, -\frac {15 \, {\left ({\left (21 \, b^{5} c^{3} - 35 \, a b^{4} c^{2} d + 15 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (21 \, a b^{4} c^{3} - 35 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (21 \, a^{2} b^{3} c^{3} - 35 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{5} c^{3} + {\left (315 \, a b^{4} c^{3} - 420 \, a^{2} b^{3} c^{2} d + 113 \, a^{3} b^{2} c d^{2}\right )} x^{4} + 2 \, {\left (210 \, a^{2} b^{3} c^{3} - 287 \, a^{3} b^{2} c^{2} d + 81 \, a^{4} b c d^{2}\right )} x^{3} + 3 \, {\left (21 \, a^{3} b^{2} c^{3} - 32 \, a^{4} b c^{2} d + 11 \, a^{5} c d^{2}\right )} x^{2} - 2 \, {\left (9 \, a^{4} b c^{3} - 13 \, a^{5} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{6} b^{2} c x^{5} + 2 \, a^{7} b c x^{4} + a^{8} 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)^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(15*((21*b^5*c^3 - 35*a*b^4*c^2*d + 15*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 + 2*(21*a*b^4*c^3 - 35*a^2*b^3*
c^2*d + 15*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + (21*a^2*b^3*c^3 - 35*a^3*b^2*c^2*d + 15*a^4*b*c*d^2 - a^5*d^3)*x^3
)*sqrt(a*c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*
x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(8*a^5*c^3 + (315*a*b^4*c^3 - 420*a^2*b^3*c^2*d + 113
*a^3*b^2*c*d^2)*x^4 + 2*(210*a^2*b^3*c^3 - 287*a^3*b^2*c^2*d + 81*a^4*b*c*d^2)*x^3 + 3*(21*a^3*b^2*c^3 - 32*a^
4*b*c^2*d + 11*a^5*c*d^2)*x^2 - 2*(9*a^4*b*c^3 - 13*a^5*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b^2*c*x^5
+ 2*a^7*b*c*x^4 + a^8*c*x^3), -1/48*(15*((21*b^5*c^3 - 35*a*b^4*c^2*d + 15*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 +
2*(21*a*b^4*c^3 - 35*a^2*b^3*c^2*d + 15*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + (21*a^2*b^3*c^3 - 35*a^3*b^2*c^2*d +
15*a^4*b*c*d^2 - a^5*d^3)*x^3)*sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x
+ c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(8*a^5*c^3 + (315*a*b^4*c^3 - 420*a^2*b^3*c^2*d + 1
13*a^3*b^2*c*d^2)*x^4 + 2*(210*a^2*b^3*c^3 - 287*a^3*b^2*c^2*d + 81*a^4*b*c*d^2)*x^3 + 3*(21*a^3*b^2*c^3 - 32*
a^4*b*c^2*d + 11*a^5*c*d^2)*x^2 - 2*(9*a^4*b*c^3 - 13*a^5*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b^2*c*x^
5 + 2*a^7*b*c*x^4 + a^8*c*x^3)]

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Sympy [F(-1)] Timed out
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)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4178 vs. \(2 (234) = 468\).
time = 46.25, size = 4178, normalized size = 15.03 \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)^(5/2),x, algorithm="giac")

[Out]

5/8*(21*sqrt(b*d)*b^3*c^3*abs(b) - 35*sqrt(b*d)*a*b^2*c^2*d*abs(b) + 15*sqrt(b*d)*a^2*b*c*d^2*abs(b) - sqrt(b*
d)*a^3*d^3*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^5*b) - 1/12*(315*sqrt(b*d)*b^19*c^11*abs(b) - 3255*sqrt(b*d)*a*b^18
*c^10*d*abs(b) + 15233*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 42597*sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b) + 79038*sqr
t(b*d)*a^4*b^15*c^7*d^4*abs(b) - 102102*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) + 93618*sqrt(b*d)*a^6*b^13*c^5*d^6*a
bs(b) - 60858*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 27447*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) - 8163*sqrt(b*d)*a^9
*b^10*c^2*d^9*abs(b) + 1437*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 113*sqrt(b*d)*a^11*b^8*d^11*abs(b) - 2520*sqrt(
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) + 19950*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d*abs(b) - 68622*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^15*c^8*d^2*abs(b) + 133080*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^14*c^7*d^3*abs(b) - 156744*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^6*d^4*abs(b) + 109956*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5*d^5*abs(b) - 37380*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6*abs(b) - 3528*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7*abs(b) + 8400*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^9*c^2*d^8*abs(b) - 2946*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^8*c*d^9*abs(b) + 354*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^7*d^10*abs(b) + 8820*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^4*b^15*c^9*abs(b) - 51870*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^4*a*b^14*c^8*d*abs(b) + 125328*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^2*b^13*c^7*d^2*abs(b) - 156264*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^3*b^12*c^6*d^3*abs(b) + 99480*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a^4*b^11*c^5*d^4*abs(b) - 20340*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^4*a^5*b^10*c^4*d^5*abs(b) - 10800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^4*a^6*b^9*c^3*d^6*abs(b) + 6552*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
4*a^7*b^8*c^2*d^7*abs(b) - 876*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^8
*b^7*c*d^8*abs(b) - 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^9*b^6*d^9
*abs(b) - 17640*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^13*c^8*abs(b) +
74550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^12*c^7*d*abs(b) - 119602
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^11*c^6*d^2*abs(b) + 86766*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^10*c^5*d^3*abs(b) - 26090*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^9*c^4*d^4*abs(b) + 7074*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^8*c^3*d^5*abs(b) - 9942*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^7*c^2*d^6*abs(b) + 6074*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^6*c*d^7*abs(b) - 1190*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^5*d^8*abs(b) + 22050*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7*abs(b) - 65100*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^10*c^6*d*abs(b) + 64638*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^9*c^5*d^2*abs(b) - 23400*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*a^3*b^8*c^4*d^3*abs(b) + 1470*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^8*a^4*b^7*c^3*d^4*abs(b) + 3828*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^5*b^6*c^2*d^5*abs(b) - 5406*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^8*a^6*b^5*c*d^6*abs(b) + 1920*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^8*a^7*b^4*d^7*abs(b) - 17640*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^10*b^9*c^6*abs(b) + 36330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^8
*c^5*d*abs(b) - 20370*sqrt(b*d)*(sqrt(b*d)*sqrt...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^4\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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