Optimal. Leaf size=119 \[ -\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}+\frac{2 \sqrt{c+d x} (b c-a d)}{a b \sqrt{a+b x}} \]
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Rubi [A] time = 0.272074, antiderivative size = 119, 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 \[ -\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}+\frac{2 \sqrt{c+d x} (b c-a d)}{a b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(x*(a + b*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 28.9037, size = 109, normalized size = 0.92 \[ \frac{2 d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}}} - \frac{2 \sqrt{c + d x} \left (a d - b c\right )}{a b \sqrt{a + b x}} - \frac{2 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/x/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.302373, size = 158, normalized size = 1.33 \[ -\frac{c^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+\frac{c^{3/2} \log (x)}{a^{3/2}}+\frac{d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}-\frac{2 \sqrt{c+d x} (a d-b c)}{a b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(x*(a + b*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.033, size = 306, normalized size = 2.6 \[{\frac{1}{ab} \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{b}^{2}{c}^{2}\sqrt{bd}+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) xab{d}^{2}\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) ab{c}^{2}\sqrt{bd}+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){a}^{2}{d}^{2}\sqrt{ac}-2\,ad\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+2\,bc\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ) \sqrt{dx+c}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/x/(b*x+a)^(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((d*x + c)^(3/2)/((b*x + a)^(3/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.701016, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/((b*x + a)^(3/2)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x\right )^{\frac{3}{2}}}{x \left (a + b x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(3/2)/x/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.585923, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/((b*x + a)^(3/2)*x),x, algorithm="giac")
[Out]