Optimal. Leaf size=200 \[ -\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}+\frac{d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\frac{e \sqrt{d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac{(d+e x)^{3/2} (c d-b e) (2 c d-b e)}{b^2 c (b+c x)}-\frac{d (d+e x)^{5/2}}{b x (b+c x)} \]
[Out]
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Rubi [A] time = 0.839595, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}+\frac{d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\frac{e \sqrt{d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac{(d+e x)^{3/2} (c d-b e) (2 c d-b e)}{b^2 c (b+c x)}-\frac{d (d+e x)^{5/2}}{b x (b+c x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(7/2)/(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 88.3359, size = 182, normalized size = 0.91 \[ - \frac{d \left (d + e x\right )^{\frac{5}{2}}}{b x \left (b + c x\right )} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{b^{2} c \left (b + c x\right )} + \frac{e \sqrt{d + e x} \left (3 b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{b^{2} c^{2}} - \frac{d^{\frac{5}{2}} \left (7 b e - 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} - \frac{\left (b e - c d\right )^{\frac{5}{2}} \left (3 b e + 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(7/2)/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.316625, size = 148, normalized size = 0.74 \[ -\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}+\frac{d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\sqrt{d+e x} \left (\frac{(b e-c d)^3}{b^2 c^2 (b+c x)}-\frac{d^3}{b^2 x}+\frac{2 e^3}{c^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^2,x]
[Out]
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Maple [B] time = 0.031, size = 403, normalized size = 2. \[ 2\,{\frac{{e}^{3}\sqrt{ex+d}}{{c}^{2}}}+{\frac{{e}^{4}b}{{c}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{{e}^{3}\sqrt{ex+d}d}{c \left ( cex+be \right ) }}+3\,{\frac{{e}^{2}\sqrt{ex+d}{d}^{2}}{b \left ( cex+be \right ) }}-{\frac{ce{d}^{3}}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{{e}^{4}b}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+5\,{\frac{{e}^{3}d}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+3\,{\frac{{d}^{2}{e}^{2}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-9\,{\frac{ce{d}^{3}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{4}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{{d}^{3}}{{b}^{2}x}\sqrt{ex+d}}-7\,{\frac{e{d}^{5/2}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{7/2}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)/(c*x^2+b*x)^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((e*x + d)^(7/2)/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.815208, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/(c*x^2 + b*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((e*x+d)**(7/2)/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.237678, size = 464, normalized size = 2.32 \[ \frac{2 \, \sqrt{x e + d} e^{3}}{c^{2}} - \frac{{\left (4 \, c d^{4} - 7 \, b d^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (4 \, c^{4} d^{4} - 9 \, b c^{3} d^{3} e + 3 \, b^{2} c^{2} d^{2} e^{2} + 5 \, b^{3} c d e^{3} - 3 \, b^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c^{2}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e - 2 \, \sqrt{x e + d} c^{3} d^{4} e - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{2} + 4 \, \sqrt{x e + d} b c^{2} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{3} - 3 \, \sqrt{x e + d} b^{2} c d^{2} e^{3} -{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{4} + \sqrt{x e + d} b^{3} d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]