Optimal. Leaf size=309 \[ -\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{64 b d^5}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{96 b d^4}+\frac{5 (b c-a d)^2 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{11/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+14 a c-\frac{63 b c^2}{d}\right )}{24 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{7/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b d^2} \]
[Out]
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Rubi [A] time = 0.751876, antiderivative size = 309, 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 \[ -\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{64 b d^5}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{96 b d^4}+\frac{5 (b c-a d)^2 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{11/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+14 a c-\frac{63 b c^2}{d}\right )}{24 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{7/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b d^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x)^(5/2))/(c + d*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 59.9697, size = 289, normalized size = 0.94 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{7}{2}}}{d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x}}{4 b d^{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right )}{24 b d^{3} \left (a d - b c\right )} - \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right )}{96 b d^{4}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right )}{64 b d^{5}} - \frac{5 \left (a d - b c\right )^{2} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**(5/2)/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.269729, size = 242, normalized size = 0.78 \[ \frac{5 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{11/2}}+\frac{\sqrt{a+b x} \left (15 a^3 d^3 (c+d x)+a^2 b d^2 \left (-839 c^2-337 c d x+118 d^2 x^2\right )+a b^2 d \left (1785 c^3+637 c^2 d x-244 c d^2 x^2+136 d^3 x^3\right )-3 b^3 \left (315 c^4+105 c^3 d x-42 c^2 d^2 x^2+24 c d^3 x^3-16 d^4 x^4\right )\right )}{192 b d^5 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x)^(5/2))/(c + d*x)^(3/2),x]
[Out]
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Maple [B] time = 0.04, size = 961, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^(5/2)/(d*x+c)^(3/2),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((b*x + a)^(5/2)*x^2/(d*x + c)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.08218, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{4} x^{4} - 945 \, b^{3} c^{4} + 1785 \, a b^{2} c^{3} d - 839 \, a^{2} b c^{2} d^{2} + 15 \, a^{3} c d^{3} - 8 \,{\left (9 \, b^{3} c d^{3} - 17 \, a b^{2} d^{4}\right )} x^{3} + 2 \,{\left (63 \, b^{3} c^{2} d^{2} - 122 \, a b^{2} c d^{3} + 59 \, a^{2} b d^{4}\right )} x^{2} -{\left (315 \, b^{3} c^{3} d - 637 \, a b^{2} c^{2} d^{2} + 337 \, a^{2} b c d^{3} - 15 \, a^{3} d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (63 \, b^{4} c^{5} - 140 \, a b^{3} c^{4} d + 90 \, a^{2} b^{2} c^{3} d^{2} - 12 \, a^{3} b c^{2} d^{3} - a^{4} c d^{4} +{\left (63 \, b^{4} c^{4} d - 140 \, a b^{3} c^{3} d^{2} + 90 \, a^{2} b^{2} c^{2} d^{3} - 12 \, a^{3} b c d^{4} - a^{4} d^{5}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \,{\left (b d^{6} x + b c d^{5}\right )} \sqrt{b d}}, \frac{2 \,{\left (48 \, b^{3} d^{4} x^{4} - 945 \, b^{3} c^{4} + 1785 \, a b^{2} c^{3} d - 839 \, a^{2} b c^{2} d^{2} + 15 \, a^{3} c d^{3} - 8 \,{\left (9 \, b^{3} c d^{3} - 17 \, a b^{2} d^{4}\right )} x^{3} + 2 \,{\left (63 \, b^{3} c^{2} d^{2} - 122 \, a b^{2} c d^{3} + 59 \, a^{2} b d^{4}\right )} x^{2} -{\left (315 \, b^{3} c^{3} d - 637 \, a b^{2} c^{2} d^{2} + 337 \, a^{2} b c d^{3} - 15 \, a^{3} d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (63 \, b^{4} c^{5} - 140 \, a b^{3} c^{4} d + 90 \, a^{2} b^{2} c^{3} d^{2} - 12 \, a^{3} b c^{2} d^{3} - a^{4} c d^{4} +{\left (63 \, b^{4} c^{4} d - 140 \, a b^{3} c^{3} d^{2} + 90 \, a^{2} b^{2} c^{2} d^{3} - 12 \, a^{3} b c d^{4} - a^{4} d^{5}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \,{\left (b d^{6} x + b c d^{5}\right )} \sqrt{-b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*x^2/(d*x + c)^(3/2),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)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.273842, size = 568, normalized size = 1.84 \[ \frac{{\left ({\left (2 \,{\left (4 \,{\left (\frac{6 \,{\left (b x + a\right )} b^{2} d^{8}}{b^{12} c d^{10} - a b^{11} d^{11}} - \frac{9 \, b^{3} c d^{7} + 7 \, a b^{2} d^{8}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )}{\left (b x + a\right )} + \frac{63 \, b^{4} c^{2} d^{6} - 14 \, a b^{3} c d^{7} - a^{2} b^{2} d^{8}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )}{\left (b x + a\right )} - \frac{5 \,{\left (63 \, b^{5} c^{3} d^{5} - 77 \, a b^{4} c^{2} d^{6} + 13 \, a^{2} b^{3} c d^{7} + a^{3} b^{2} d^{8}\right )}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (63 \, b^{6} c^{4} d^{4} - 140 \, a b^{5} c^{3} d^{5} + 90 \, a^{2} b^{4} c^{2} d^{6} - 12 \, a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )} \sqrt{b x + a}}{8257536 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{5 \,{\left (63 \, b^{3} c^{3} - 77 \, a b^{2} c^{2} d + 13 \, a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{2752512 \, \sqrt{b d} b^{9} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*x^2/(d*x + c)^(3/2),x, algorithm="giac")
[Out]