3.783 \(\int \frac{1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=268 \[ \frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right ) (a d+b c)}{4 a^3 c^3 \sqrt{c+d x} (b c-a d)^2}+\frac{b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{4 a^3 c^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}-\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{7/2}}+\frac{5 (a d+b c)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}} \]

[Out]

(b*(15*b^2*c^2 - 2*a*b*c*d - 5*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - 1/(2*a*c*x^2*Sq
rt[a + b*x]*Sqrt[c + d*x]) + (5*(b*c + a*d))/(4*a^2*c^2*x*Sqrt[a + b*x]*Sqrt[c + d*x]) + (d*(b*c + a*d)*(15*b^
2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x])/(4*a^3*c^3*(b*c - a*d)^2*Sqrt[c + d*x]) - (3*(5*b^2*c^2 + 6*a*
b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.25839, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {103, 151, 152, 12, 93, 208} \[ \frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right ) (a d+b c)}{4 a^3 c^3 \sqrt{c+d x} (b c-a d)^2}+\frac{b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{4 a^3 c^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}-\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{7/2}}+\frac{5 (a d+b c)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(b*(15*b^2*c^2 - 2*a*b*c*d - 5*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - 1/(2*a*c*x^2*Sq
rt[a + b*x]*Sqrt[c + d*x]) + (5*(b*c + a*d))/(4*a^2*c^2*x*Sqrt[a + b*x]*Sqrt[c + d*x]) + (d*(b*c + a*d)*(15*b^
2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x])/(4*a^3*c^3*(b*c - a*d)^2*Sqrt[c + d*x]) - (3*(5*b^2*c^2 + 6*a*
b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(7/2)*c^(7/2))

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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 152

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 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{1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}-\frac{\int \frac{\frac{5}{2} (b c+a d)+3 b d x}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{2 a c}\\ &=-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{5 (b c+a d)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}+\frac{\int \frac{\frac{3}{4} \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )+5 b d (b c+a d) x}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{2 a^2 c^2}\\ &=\frac{b \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{5 (b c+a d)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}+\frac{\int \frac{\frac{3}{8} (b c-a d) \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )+\frac{1}{4} b d \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right ) x}{x \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{a^3 c^2 (b c-a d)}\\ &=\frac{b \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{5 (b c+a d)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}+\frac{d (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x}}{4 a^3 c^3 (b c-a d)^2 \sqrt{c+d x}}-\frac{2 \int -\frac{3 (b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{a^3 c^3 (b c-a d)^2}\\ &=\frac{b \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{5 (b c+a d)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}+\frac{d (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x}}{4 a^3 c^3 (b c-a d)^2 \sqrt{c+d x}}+\frac{\left (3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 a^3 c^3}\\ &=\frac{b \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{5 (b c+a d)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}+\frac{d (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x}}{4 a^3 c^3 (b c-a d)^2 \sqrt{c+d x}}+\frac{\left (3 \left (5 b^2 c^2+6 a b c d+5 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 )}{4 a^3 c^3}\\ &=\frac{b \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{5 (b c+a d)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}+\frac{d (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x}}{4 a^3 c^3 (b c-a d)^2 \sqrt{c+d x}}-\frac{3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.263429, size = 254, normalized size = 0.95 \[ -\frac{a^2 b^2 c \left (5 c^2 d x+2 c^3+10 c d^2 x^2+7 d^3 x^3\right )+a^3 b d \left (5 c^2 d x-4 c^3+2 c d^2 x^2-15 d^3 x^3\right )+a^4 d^2 \left (2 c^2-5 c d x-15 d^2 x^2\right )+a b^3 c^2 x \left (-5 c^2+2 c d x+7 d^2 x^2\right )-15 b^4 c^3 x^2 (c+d x)}{4 a^3 c^3 x^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

-(-15*b^4*c^3*x^2*(c + d*x) + a^4*d^2*(2*c^2 - 5*c*d*x - 15*d^2*x^2) + a*b^3*c^2*x*(-5*c^2 + 2*c*d*x + 7*d^2*x
^2) + a^3*b*d*(-4*c^3 + 5*c^2*d*x + 2*c*d^2*x^2 - 15*d^3*x^3) + a^2*b^2*c*(2*c^3 + 5*c^2*d*x + 10*c*d^2*x^2 +
7*d^3*x^3))/(4*a^3*c^3*(b*c - a*d)^2*x^2*Sqrt[a + b*x]*Sqrt[c + d*x]) - (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)
*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(7/2)*c^(7/2))

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Maple [B]  time = 0.04, size = 1372, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a^3/c^3*(14*x^3*a^2*b^2*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+14*x^3*a*b^3*c^2*d^2*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2)+4*x^2*a^3*b*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+20*x^2*a^2*b^2*c^2*d^2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+4*x^2*a*b^3*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10*x*a^3*b*c^2*d^2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+10*x*a^2*b^2*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^2*b^4*c^4*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+4*a^4*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a^2*b^2*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)+15*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*b*d^5+15*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^5*c^4*d+15*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^2*a^5*c*d^4+15*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^2*a*b^4*c^5-
30*x^2*a^4*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+3*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^3*a^4*b*c*d^4+15*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^3*a^5*d^5+15*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^3*b^5*c^5-18*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^3*a^3*b^2*c^2*d^3-18*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^3*a^2*b^3*c^3*d^2+3*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^3*a*b^4*c^4*d-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^2*a^4*b*c^2*d^3-6*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^2*a^3*b^2*c^3*d^2-12*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^2*a^2*b^3*c^4*d-30*x^3*a^3*b*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^3*b^4*c^3*d*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)-10*x*a^4*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10*x*a*b^3*c^4*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)-8*a^3*b*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-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^2*c*d^4-6*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^3*c^2*d^3-12*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*b^4*c^3*d^2)/((
b*x+a)*(d*x+c))^(1/2)/(a*d-b*c)^2/(a*c)^(1/2)/x^2/(b*x+a)^(1/2)/(d*x+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(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 29.0625, size = 2511, normalized size = 9.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(3*((5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^4 + (5*b^5*c^5
+ a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 4*a^2*b
^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x^2)*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*(2*a^3*b^2*c^5 - 4*a^4*b*c^4*d + 2*a^5*c^3*d^2 - (15*a*b^4*c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c
^2*d^3 + 15*a^4*b*c*d^4)*x^3 - (15*a*b^4*c^5 - 2*a^2*b^3*c^4*d - 10*a^3*b^2*c^3*d^2 - 2*a^4*b*c^2*d^3 + 15*a^5
*c*d^4)*x^2 - 5*(a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((
a^4*b^3*c^6*d - 2*a^5*b^2*c^5*d^2 + a^6*b*c^4*d^3)*x^4 + (a^4*b^3*c^7 - a^5*b^2*c^6*d - a^6*b*c^5*d^2 + a^7*c^
4*d^3)*x^3 + (a^5*b^2*c^7 - 2*a^6*b*c^6*d + a^7*c^5*d^2)*x^2), 1/8*(3*((5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*
b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^4 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^
2*d^3 + a^4*b*c*d^4 + 5*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2*d^3 +
5*a^5*c*d^4)*x^2)*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*(2*a^3*b^2*c^5 - 4*a^4*b*c^4*d + 2*a^5*c^3*d^2 - (15*a*b^4*c^4*d
- 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4)*x^3 - (15*a*b^4*c^5 - 2*a^2*b^3*c^4*d - 10*a^3*b^2*
c^3*d^2 - 2*a^4*b*c^2*d^3 + 15*a^5*c*d^4)*x^2 - 5*(a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*
x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^4*b^3*c^6*d - 2*a^5*b^2*c^5*d^2 + a^6*b*c^4*d^3)*x^4 + (a^4*b^3*c^7 - a^5*
b^2*c^6*d - a^6*b*c^5*d^2 + a^7*c^4*d^3)*x^3 + (a^5*b^2*c^7 - 2*a^6*b*c^6*d + a^7*c^5*d^2)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError