Optimal. Leaf size=163 \[ \frac{c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{\sqrt{c+d x} (3 b c-2 a d) (b c-a d)}{a^2 b \sqrt{a+b x}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c (c+d x)^{3/2}}{a x \sqrt{a+b x}} \]
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Rubi [A] time = 0.522031, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{\sqrt{c+d x} (3 b c-2 a d) (b c-a d)}{a^2 b \sqrt{a+b x}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c (c+d x)^{3/2}}{a x \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(5/2)/(x^2*(a + b*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 32.5743, size = 150, normalized size = 0.92 \[ \frac{2 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}}} - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{a x \sqrt{a + b x}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (2 a d - 3 b c\right )}{a^{2} b \sqrt{a + b x}} - \frac{c^{\frac{3}{2}} \left (5 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)/x**2/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.543031, size = 199, normalized size = 1.22 \[ \frac{1}{2} \left (\frac{c^{3/2} \log (x) (5 a d-3 b c)}{a^{5/2}}+\frac{c^{3/2} (3 b c-5 a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 (b c-a d)^2}{b (a+b x)}+\frac{c^2}{x}\right )}{a^2}+\frac{2 d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(5/2)/(x^2*(a + b*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.037, size = 502, normalized size = 3.1 \[ -{\frac{1}{2\,{a}^{2}xb}\sqrt{dx+c} \left ( 5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{3}{c}^{3}\sqrt{bd}-2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}b{d}^{3}\sqrt{ac}+5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}b{c}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xa{b}^{2}{c}^{3}\sqrt{bd}-2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}{d}^{3}\sqrt{ac}+4\,x{a}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-8\,xabcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+6\,x{b}^{2}{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+2\,ab{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)/x^2/(b*x+a)^(3/2),x)
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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)^(5/2)/((b*x + a)^(3/2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.71938, 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)^(5/2)/((b*x + a)^(3/2)*x^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((d*x+c)**(5/2)/x**2/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.630736, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)^(3/2)*x^2),x, algorithm="giac")
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