3.791 \(\int \frac{1}{x^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{d \sqrt{a+b x} \left (31 a^2 b c d^2-15 a^3 d^3-9 a b^2 c^2 d+9 b^3 c^3\right )}{3 a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{d \sqrt{a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}+\frac{(5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{7/2}}-\frac{b (3 b c-a d)}{a^2 c \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}} \]

[Out]

-((b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))) - 1/(a*c*x*Sqrt[a + b*x]*(c + d*x)^(3/2
)) - (d*(9*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (d*(9*b
^3*c^3 - 9*a*b^2*c^2*d + 31*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x])/(3*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) +
((3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(5/2)*c^(7/2))

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Rubi [A]  time = 0.274278, antiderivative size = 264, 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 = {103, 152, 12, 93, 208} \[ -\frac{d \sqrt{a+b x} \left (31 a^2 b c d^2-15 a^3 d^3-9 a b^2 c^2 d+9 b^3 c^3\right )}{3 a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{d \sqrt{a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}+\frac{(5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{7/2}}-\frac{b (3 b c-a d)}{a^2 c \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))) - 1/(a*c*x*Sqrt[a + b*x]*(c + d*x)^(3/2
)) - (d*(9*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (d*(9*b
^3*c^3 - 9*a*b^2*c^2*d + 31*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x])/(3*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) +
((3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(5/2)*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 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^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{\int \frac{\frac{1}{2} (3 b c+5 a d)+3 b d x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{a c}\\ &=-\frac{b (3 b c-a d)}{a^2 c (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{2 \int \frac{\frac{1}{4} (b c-a d) (3 b c+5 a d)+b d (3 b c-a d) x}{x \sqrt{a+b x} (c+d x)^{5/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac{b (3 b c-a d)}{a^2 c (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}+\frac{4 \int \frac{-\frac{3}{8} (b c-a d)^2 (3 b c+5 a d)-\frac{1}{4} b d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac{b (3 b c-a d)}{a^2 c (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}-\frac{8 \int \frac{3 (b c-a d)^3 (3 b c+5 a d)}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac{b (3 b c-a d)}{a^2 c (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}-\frac{(3 b c+5 a d) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a^2 c^3}\\ &=-\frac{b (3 b c-a d)}{a^2 c (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}-\frac{(3 b c+5 a d) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{a^2 c^3}\\ &=-\frac{b (3 b c-a d)}{a^2 c (b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac{d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt{c+d x}}+\frac{(3 b c+5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.553921, size = 263, normalized size = 1. \[ \frac{-\frac{d \sqrt{a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{a c (b c-a d)^2}+\frac{(c+d x) \left (\sqrt{a} \sqrt{c} d \sqrt{a+b x} \left (-31 a^2 b c d^2+15 a^3 d^3+9 a b^2 c^2 d-9 b^3 c^3\right )+3 \sqrt{c+d x} (5 a d+3 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )}{a^{3/2} c^{5/2} (b c-a d)^3}-\frac{3 b (a d-3 b c)}{a \sqrt{a+b x} (a d-b c)}-\frac{3}{x \sqrt{a+b x}}}{3 a c (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b*(-3*b*c + a*d))/(a*(-(b*c) + a*d)*Sqrt[a + b*x]) - 3/(x*Sqrt[a + b*x]) - (d*(9*b^2*c^2 - 6*a*b*c*d + 5*
a^2*d^2)*Sqrt[a + b*x])/(a*c*(b*c - a*d)^2) + ((c + d*x)*(Sqrt[a]*Sqrt[c]*d*(-9*b^3*c^3 + 9*a*b^2*c^2*d - 31*a
^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[a + b*x] + 3*(b*c - a*d)^3*(3*b*c + 5*a*d)*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a
+ b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(3/2)*c^(5/2)*(b*c - a*d)^3))/(3*a*c*(c + d*x)^(3/2))

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Maple [B]  time = 0.046, size = 1692, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(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*a^5*c^2*d^4-9*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*a*b^4*c^6+36*x^2*b^4*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-40*
x*a^4*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+18*a^3*b*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-18*a^2*b^
2*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-36*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^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^4*a^2*b^3*c^2*d^4+12*l
n((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^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^3*a^4*b*c*d^5-54*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^3*a^3*b^2*c^2*d^4+48*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^3+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*b^4*c^4*d^2-57*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^4+42*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^2-6*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*b^4*c^5*d-36*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*a^4*b*c^3*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*a^3*b^2*c^4*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*a^2*b^3*c^5*d-30*x^3*a^3*b*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-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*b^5*c^5*d+62*x^3*a^2*b^2*c*d^4*((b*x+a)*(d*x+c))^
(1/2)*(a*c)^(1/2)-18*x^3*a*b^3*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+22*x^2*a^3*b*c*d^4*((b*x+a)*(d*x+c)
)^(1/2)*(a*c)^(1/2)+66*x^2*a^2*b^2*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30*x^2*a*b^3*c^3*d^2*((b*x+a)*(
d*x+c))^(1/2)*(a*c)^(1/2)+78*x*a^3*b*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-18*x*a^2*b^2*c^3*d^2*((b*x+a)
*(d*x+c))^(1/2)*(a*c)^(1/2)-6*x*a*b^3*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+18*x^3*b^4*c^3*d^2*((b*x+a)*(d
*x+c))^(1/2)*(a*c)^(1/2)+30*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^5-30*x
^2*a^4*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+18*x*b^4*c^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6*a^4*c^2*d^3*
((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+6*a*b^3*c^5*((b*x+a)*(d*x+c))^(1/2)*(a*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^6-9*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*b^5*c^4*d^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^3*a^5*d^6-9*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^2*b^5*c^6)/c^3/a^2/x/(a*c)^(1/2)/(a*d-b*c)^3/(
(b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 33.983, size = 3363, normalized size = 12.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((3*b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 - 6*a^2*b^3*c^2*d^4 + 12*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (6*b^5*
c^5*d - 5*a*b^4*c^4*d^2 - 16*a^2*b^3*c^3*d^3 + 18*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (3*b^5*c^
6 + 2*a*b^4*c^5*d - 14*a^2*b^3*c^4*d^2 + 19*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (3*a*b^4*c^6 - 4*a^2*b^3*c^5*d
- 6*a^3*b^2*c^4*d^2 + 12*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*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*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + (9*a*b^4*c^4*d^2 - 9*a^2*b^3*c^3*d^3
+ 31*a^3*b^2*c^2*d^4 - 15*a^4*b*c*d^5)*x^3 + (18*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 33*a^3*b^2*c^3*d^3 + 11*a
^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (9*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 39*a^4*b*c^3*d^3 - 20*
a^5*c^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b
*c^4*d^5)*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 +
(a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5
*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x), -1/6*(3*((3*b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 - 6*a^2*b^3*c^2*d^4
+ 12*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (6*b^5*c^5*d - 5*a*b^4*c^4*d^2 - 16*a^2*b^3*c^3*d^3 + 18*a^3*b^2*c^2*d
^4 + 2*a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (3*b^5*c^6 + 2*a*b^4*c^5*d - 14*a^2*b^3*c^4*d^2 + 19*a^4*b*c^2*d^4 - 10*
a^5*c*d^5)*x^2 + (3*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 12*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqr
t(-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*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + (9*a*b^4*c^4*d^2
- 9*a^2*b^3*c^3*d^3 + 31*a^3*b^2*c^2*d^4 - 15*a^4*b*c*d^5)*x^3 + (18*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 33*a^
3*b^2*c^3*d^3 + 11*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (9*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 39
*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5
*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 -
a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 +
(a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError