Optimal. Leaf size=124 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac{5 b d}{\sqrt{c+d x} (b c-a d)^3}-\frac{1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \]
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Rubi [A] time = 0.149603, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac{5 b d}{\sqrt{c+d x} (b c-a d)^3}-\frac{1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^2*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 27.5669, size = 109, normalized size = 0.88 \[ \frac{5 b^{\frac{3}{2}} d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\left (a d - b c\right )^{\frac{7}{2}}} + \frac{5 b d}{\sqrt{c + d x} \left (a d - b c\right )^{3}} - \frac{5 d}{3 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**2/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.24182, size = 125, normalized size = 1.01 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}+\sqrt{c+d x} \left (-\frac{b^2}{(a+b x) (b c-a d)^3}-\frac{4 b d}{(c+d x) (b c-a d)^3}-\frac{2 d}{3 (c+d x)^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^2*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.027, size = 125, normalized size = 1. \[ -{\frac{2\,d}{3\, \left ( ad-bc \right ) ^{2}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{bd}{ \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}+{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+5\,{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^2/(d*x+c)^(5/2),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(1/((b*x + a)^2*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227061, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 28 \, a b c d - 4 \, a^{2} d^{2} + 15 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) + 20 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{6 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \sqrt{d x + c}}, -\frac{15 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} - 15 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x + c} b}\right ) + 10 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{3 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \sqrt{d x + c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**2/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220147, size = 292, normalized size = 2.35 \[ -\frac{5 \, b^{2} d \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x + c} b^{2} d}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} - \frac{2 \,{\left (6 \,{\left (d x + c\right )} b d + b c d - a d^{2}\right )}}{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]