Optimal. Leaf size=186 \[ \frac{4 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b \sqrt{c+d x} (b c-a d)^4}+\frac{2 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b^2 (c+d x)^{3/2} (b c-a d)^3}-\frac{2 a^2}{3 b^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac{4 a c}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.439923, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{4 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b \sqrt{c+d x} (b c-a d)^4}+\frac{2 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b^2 (c+d x)^{3/2} (b c-a d)^3}-\frac{2 a^2}{3 b^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac{4 a c}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 37.7661, size = 172, normalized size = 0.92 \[ \frac{4 a c}{d \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 c^{2}}{3 d^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 \sqrt{c + d x} \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{3 d \sqrt{a + b x} \left (a d - b c\right )^{4}} + \frac{2 \sqrt{c + d x} \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{3 d^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.329903, size = 107, normalized size = 0.58 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2 (a d-b c)}{(a+b x)^2}+\frac{c^2 (b c-a d)}{(c+d x)^2}+\frac{2 a (a d+3 b c)}{a+b x}+\frac{2 c (3 a d+b c)}{c+d x}\right )}{3 (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.013, size = 203, normalized size = 1.1 \[{\frac{4\,{a}^{2}b{d}^{3}{x}^{3}+24\,a{b}^{2}c{d}^{2}{x}^{3}+4\,{b}^{3}{c}^{2}d{x}^{3}+6\,{a}^{3}{d}^{3}{x}^{2}+42\,{a}^{2}bc{d}^{2}{x}^{2}+42\,a{b}^{2}{c}^{2}d{x}^{2}+6\,{b}^{3}{c}^{3}{x}^{2}+24\,{a}^{3}c{d}^{2}x+48\,{a}^{2}b{c}^{2}dx+24\,a{b}^{2}{c}^{3}x+16\,{a}^{3}{c}^{2}d+16\,{a}^{2}b{c}^{3}}{3\,{a}^{4}{d}^{4}-12\,{a}^{3}bc{d}^{3}+18\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-12\,a{b}^{3}{c}^{3}d+3\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.754237, size = 632, normalized size = 3.4 \[ \frac{2 \,{\left (8 \, a^{2} b c^{3} + 8 \, a^{3} c^{2} d + 2 \,{\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \,{\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 12 \,{\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.505282, size = 852, normalized size = 4.58 \[ -\frac{\sqrt{b x + a}{\left (\frac{2 \,{\left (b^{7} c^{5} d^{2}{\left | b \right |} - 6 \, a^{2} b^{5} c^{3} d^{4}{\left | b \right |} + 8 \, a^{3} b^{4} c^{2} d^{5}{\left | b \right |} - 3 \, a^{4} b^{3} c d^{6}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{8} c^{6} d{\left | b \right |} - 2 \, a b^{7} c^{5} d^{2}{\left | b \right |} - 2 \, a^{2} b^{6} c^{4} d^{3}{\left | b \right |} + 8 \, a^{3} b^{5} c^{3} d^{4}{\left | b \right |} - 7 \, a^{4} b^{4} c^{2} d^{5}{\left | b \right |} + 2 \, a^{5} b^{3} c d^{6}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{8 \,{\left (3 \, \sqrt{b d} a b^{6} c^{3} - 5 \, \sqrt{b d} a^{2} b^{5} c^{2} d + \sqrt{b d} a^{3} b^{4} c d^{2} + \sqrt{b d} a^{4} b^{3} d^{3} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c d + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} d^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{2} c\right )}}{3 \,{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]