3.615 \(\int \frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=273 \[ \frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{7/2}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]

[Out]

(-3*(b*c - a*d)^3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^3*c^3*x) + ((b*c - a*d)^2*(b*c + a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(64*a^2*c^3*x^2) + ((b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(16*a*c^3*
x^3) + ((b*c + a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*a*c^2*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*a*c*
x^5) + (3*(b*c - a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(
7/2))

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Rubi [A]  time = 0.149952, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ \frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{7/2}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

(-3*(b*c - a*d)^3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^3*c^3*x) + ((b*c - a*d)^2*(b*c + a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(64*a^2*c^3*x^2) + ((b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(16*a*c^3*
x^3) + ((b*c + a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*a*c^2*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*a*c*
x^5) + (3*(b*c - a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(
7/2))

Rule 96

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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{(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx &=-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac{(b c+a d) \int \frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x^5} \, dx}{2 a c}\\ &=\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac{\left (3 \left (b^2-\frac{a^2 d^2}{c^2}\right )\right ) \int \frac{\sqrt{a+b x} (c+d x)^{3/2}}{x^4} \, dx}{16 a}\\ &=\frac{\left (b^2-\frac{a^2 d^2}{c^2}\right ) \sqrt{a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac{\left ((b c-a d)^2 (b c+a d)\right ) \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{32 a c^3}\\ &=\frac{(b c-a d)^2 (b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac{\left (b^2-\frac{a^2 d^2}{c^2}\right ) \sqrt{a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac{\left (3 (b c-a d)^3 (b c+a d)\right ) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{128 a^2 c^3}\\ &=-\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 a^3 c^3 x}+\frac{(b c-a d)^2 (b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac{\left (b^2-\frac{a^2 d^2}{c^2}\right ) \sqrt{a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac{\left (3 (b c-a d)^4 (b c+a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 a^3 c^3}\\ &=-\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 a^3 c^3 x}+\frac{(b c-a d)^2 (b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac{\left (b^2-\frac{a^2 d^2}{c^2}\right ) \sqrt{a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}-\frac{\left (3 (b c-a d)^4 (b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{128 a^3 c^3}\\ &=-\frac{3 (b c-a d)^3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{128 a^3 c^3 x}+\frac{(b c-a d)^2 (b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac{\left (b^2-\frac{a^2 d^2}{c^2}\right ) \sqrt{a+b x} (c+d x)^{5/2}}{16 a c x^3}+\frac{(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac{3 (b c-a d)^4 (b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.598506, size = 228, normalized size = 0.84 \[ \frac{(a d+b c) \left (16 a^{5/2} c^{3/2} (a+b x)^{3/2} (c+d x)^{5/2}+x (b c-a d) \left (8 a^{5/2} \sqrt{c} \sqrt{a+b x} (c+d x)^{5/2}+x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )\right )\right )}{128 a^{7/2} c^{7/2} x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

-((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*a*c*x^5) + ((b*c + a*d)*(16*a^(5/2)*c^(3/2)*(a + b*x)^(3/2)*(c + d*x)^(5
/2) + (b*c - a*d)*x*(8*a^(5/2)*Sqrt[c]*Sqrt[a + b*x]*(c + d*x)^(5/2) + (b*c - a*d)*x*(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])]))))/(128*a^(7/2)*c^(7/2)*x^4)

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Maple [B]  time = 0.018, size = 967, normalized size = 3.5 \begin{align*}{\frac{1}{1280\,{a}^{3}{c}^{3}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+80\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-36\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-52\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-52\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}-544\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}-352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d-352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}-256\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x)

[Out]

1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2
*a*c)/x)*x^5*a^5*d^5-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a^4*b*c*d^
4+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a^3*b^2*c^2*d^3+30*ln((a*d*x+
b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a^2*b^3*c^3*d^2-45*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a*b^4*c^4*d+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*b^5*c^5-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*d^4+80*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b*c*d^3-36*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^2
*c^2*d^2+80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^3*c^3*d-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*x^4*b^4*c^4+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^4*c*d^3-52*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*x^3*a^3*b*c^2*d^2-52*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^2*c^3*d+20*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^4*c^
2*d^2-544*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b*c^3*d-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*x^2*a^2*b^2*c^4-352*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d-352*(a*c)^(1/2)*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*x*a^3*b*c^4-256*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^4*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)/x^5/(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((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 78.6661, size = 1558, normalized size = 5.71 \begin{align*} \left [\frac{15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt{a c} x^{5} \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 (128 \, a^{5} c^{5} +{\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \,{\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \,{\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \,{\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, a^{4} c^{4} x^{5}}, -\frac{15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt{-a c} x^{5} \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 (128 \, a^{5} c^{5} +{\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \,{\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \,{\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \,{\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, a^{4} c^{4} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[1/2560*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*sqrt(a
*c)*x^5*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*(128*a^5*c^5 + (15*a*b^4*c^5 - 40*a^2*b^3*c^4*d + 18*a^3*
b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 15*a^5*c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 13*a^3*b^2*c^4*d - 13*a^4*b*c^3*d^2 +
5*a^5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^5 + 34*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 176*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(
b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^5), -1/1280*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^
2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sq
rt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(128*a^5*c^5 + (15*a*b^4*c^5 - 40*a^2*b^3*c^4
*d + 18*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 15*a^5*c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 13*a^3*b^2*c^4*d - 13*a^4*
b*c^3*d^2 + 5*a^5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^5 + 34*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 176*(a^4*b*c^5 + a^5*c^4
*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(d*x+c)**(3/2)/x**6,x)

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/2)/x**6, 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((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError