3.713 \(\int \frac{(c+d x)^{3/2}}{x^5 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=266 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{96 a^3 c x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 b c d^2+9 a^3 d^3-145 a b^2 c^2 d+105 b^3 c^3\right )}{192 a^4 c^2 x}-\frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (7 b c-9 a d)}{24 a^2 x^3}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4} \]

[Out]

-(c*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a*x^4) + ((7*b*c - 9*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^2*x^3) - ((35
*b^2*c^2 - 46*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^3*c*x^2) + ((105*b^3*c^3 - 145*a*b^2*c^2
*d + 15*a^2*b*c*d^2 + 9*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^4*c^2*x) - ((b*c - a*d)^2*(35*b^2*c^2 + 1
0*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(9/2)*c^(5/2))

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Rubi [A]  time = 0.207194, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {98, 151, 12, 93, 208} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{96 a^3 c x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 b c d^2+9 a^3 d^3-145 a b^2 c^2 d+105 b^3 c^3\right )}{192 a^4 c^2 x}-\frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (7 b c-9 a d)}{24 a^2 x^3}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(x^5*Sqrt[a + b*x]),x]

[Out]

-(c*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a*x^4) + ((7*b*c - 9*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a^2*x^3) - ((35
*b^2*c^2 - 46*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^3*c*x^2) + ((105*b^3*c^3 - 145*a*b^2*c^2
*d + 15*a^2*b*c*d^2 + 9*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^4*c^2*x) - ((b*c - a*d)^2*(35*b^2*c^2 + 1
0*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(9/2)*c^(5/2))

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 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 12

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

Rule 93

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 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)^{3/2}}{x^5 \sqrt{a+b x}} \, dx &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}-\frac{\int \frac{\frac{1}{2} c (7 b c-9 a d)+d (3 b c-4 a d) x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{4 a}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}+\frac{(7 b c-9 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^2 x^3}+\frac{\int \frac{\frac{1}{4} c \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right )+b c d (7 b c-9 a d) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{12 a^2 c}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}+\frac{(7 b c-9 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^2 x^3}-\frac{\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^3 c x^2}-\frac{\int \frac{\frac{1}{8} c \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right )+\frac{1}{4} b c d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^3 c^2}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}+\frac{(7 b c-9 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^2 x^3}-\frac{\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^3 c x^2}+\frac{\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^4 c^2 x}+\frac{\int \frac{3 c (b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^4 c^3}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}+\frac{(7 b c-9 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^2 x^3}-\frac{\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^3 c x^2}+\frac{\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^4 c^2 x}+\frac{\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^4 c^2}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}+\frac{(7 b c-9 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^2 x^3}-\frac{\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^3 c x^2}+\frac{\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^4 c^2 x}+\frac{\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 a^4 c^2}\\ &=-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4}+\frac{(7 b c-9 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a^2 x^3}-\frac{\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^3 c x^2}+\frac{\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^4 c^2 x}-\frac{(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.305998, size = 193, normalized size = 0.73 \[ -\frac{\frac{x^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}+48 a c \sqrt{a+b x} (c+d x)^{5/2}-8 x \sqrt{a+b x} (c+d x)^{5/2} (3 a d+7 b c)}{192 a^2 c^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/(x^5*Sqrt[a + b*x]),x]

[Out]

-(48*a*c*Sqrt[a + b*x]*(c + d*x)^(5/2) - 8*(7*b*c + 3*a*d)*x*Sqrt[a + b*x]*(c + d*x)^(5/2) + ((35*b^2*c^2 + 10
*a*b*c*d + 3*a^2*d^2)*x^2*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*(b*c -
a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(5/2)*Sqrt[c]))/(192*a^2*c^2*x^4)

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Maple [B]  time = 0.023, size = 593, normalized size = 2.2 \begin{align*} -{\frac{1}{384\,{a}^{4}{c}^{2}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}+12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}+54\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-180\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-18\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{3}{d}^{3}-30\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{2}bc{d}^{2}+290\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}a{b}^{2}{c}^{2}d-210\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{b}^{3}{c}^{3}+12\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{3}c{d}^{2}-184\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{2}b{c}^{2}d+140\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}a{b}^{2}{c}^{3}+144\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{3}{c}^{2}d-112\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{c}^{3}+96\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{c}^{3} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x)

[Out]

-1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^4/c^2*(9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*
x^4*a^4*d^4+12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^3*b*c*d^3+54*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b^2*c^2*d^2-180*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a*b^3*c^3*d+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)
/x)*x^4*b^4*c^4-18*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^3*d^3-30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*
a^2*b*c*d^2+290*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b^2*c^2*d-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^
3*b^3*c^3+12*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^3*c*d^2-184*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^2
*b*c^2*d+140*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a*b^2*c^3+144*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^3*c
^2*d-112*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b*c^3+96*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c^3)/((b*x
+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 24.3384, size = 1268, normalized size = 4.77 \begin{align*} \left [\frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt{a c} x^{4} \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 (48 \, a^{4} c^{4} -{\left (105 \, a b^{3} c^{4} - 145 \, a^{2} b^{2} c^{3} d + 15 \, a^{3} b c^{2} d^{2} + 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (35 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{3} b c^{4} - 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{5} c^{3} x^{4}}, \frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt{-a c} x^{4} \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 (48 \, a^{4} c^{4} -{\left (105 \, a b^{3} c^{4} - 145 \, a^{2} b^{2} c^{3} d + 15 \, a^{3} b c^{2} d^{2} + 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (35 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{3} b c^{4} - 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{5} c^{3} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/768*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 - (105*a*b^3*c^4 - 145*a^2*b^2*c^3*d + 15*a^3*b*c^2*d^2 + 9*
a^4*c*d^3)*x^3 + 2*(35*a^2*b^2*c^4 - 46*a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 - 8*(7*a^3*b*c^4 - 9*a^4*c^3*d)*x)*sq
rt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^4), 1/384*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b
*c*d^3 + 3*a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 - (105*a*b^3*c^4 - 145*a^2*b^2*c^3*d + 15*a^3*
b*c^2*d^2 + 9*a^4*c*d^3)*x^3 + 2*(35*a^2*b^2*c^4 - 46*a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 - 8*(7*a^3*b*c^4 - 9*a^
4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2)/x**5/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError