Optimal. Leaf size=248 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}-\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+10 a c-\frac{35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{5/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b d^2} \]
[Out]
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Rubi [A] time = 0.596707, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}-\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+10 a c-\frac{35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{5/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b d^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x)^(3/2))/(c + d*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 44.1898, size = 228, normalized size = 0.92 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{5}{2}}}{d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3 b d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right )}{12 b d^{3} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right )}{8 b d^{4}} - \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{3}{2}} d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**(3/2)/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.19361, size = 189, normalized size = 0.76 \[ \frac{\sqrt{a+b x} \left (3 a^2 d^2 (c+d x)+2 a b d \left (-50 c^2-19 c d x+7 d^2 x^2\right )+b^2 \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )}{24 b d^4 \sqrt{c+d x}}-\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{3/2} d^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x)^(3/2))/(c + d*x)^(3/2),x]
[Out]
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Maple [B] time = 0.038, size = 692, normalized size = 2.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^(3/2)/(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)^(3/2)*x^2/(d*x + c)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.597564, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{3} x^{3} + 105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2} - 14 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} +{\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \,{\left (b d^{5} x + b c d^{4}\right )} \sqrt{b d}}, \frac{2 \,{\left (8 \, b^{2} d^{3} x^{3} + 105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2} - 14 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} +{\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \,{\left (b d^{5} x + b c d^{4}\right )} \sqrt{-b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*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)**(3/2)/(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.258565, size = 428, normalized size = 1.73 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}} - \frac{7 \, b^{3} c d^{5} + 5 \, a b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{35 \, b^{4} c^{2} d^{4} - 10 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{3 \,{\left (35 \, b^{5} c^{3} d^{3} - 45 \, a b^{4} c^{2} d^{4} + 9 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{{\left (35 \, b^{2} c^{2} - 10 \, a b c d - a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{61440 \, \sqrt{b d} b^{7} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^2/(d*x + c)^(3/2),x, algorithm="giac")
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