Optimal. Leaf size=126 \[ -\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} \sqrt{d}}+\frac{4 a \sqrt{c+d x} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.235581, antiderivative size = 126, 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 \[ -\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} \sqrt{d}}+\frac{4 a \sqrt{c+d x} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 22.7368, size = 117, normalized size = 0.93 \[ \frac{2 a^{2} \sqrt{c + d x}}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{4 a \sqrt{c + d x} \left (2 a d - 3 b c\right )}{3 b^{2} \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**(5/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.240396, size = 116, normalized size = 0.92 \[ \frac{2 a \sqrt{c+d x} \left (-3 a^2 d+a b (5 c-4 d x)+6 b^2 c x\right )}{3 b^2 (a+b x)^{3/2} (b c-a d)^2}+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2} \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]
[Out]
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Maple [B] time = 0.036, size = 604, normalized size = 4.8 \[{\frac{1}{3\, \left ( ad-bc \right ) ^{2}{b}^{2}}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{b}^{2}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{3}cd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{4}{c}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}b{d}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}{b}^{2}cd+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{3}{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bcd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}-8\,x{a}^{2}bd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+12\,xa{b}^{2}c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,{a}^{3}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+10\,{a}^{2}bc\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^(5/2)/(d*x+c)^(1/2),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)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.411298, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (5 \, a^{2} b c - 3 \, a^{3} d + 2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )} \sqrt{b d}}, \frac{2 \,{\left (5 \, a^{2} b c - 3 \, a^{3} d + 2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )} \sqrt{-b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(5/2)*sqrt(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**(5/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.571745, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(5/2)*sqrt(d*x + c)),x, algorithm="giac")
[Out]