Optimal. Leaf size=167 \[ -\frac{35 b^{3/2} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{9/2}}+\frac{35 b d^2}{4 \sqrt{c+d x} (b c-a d)^4}+\frac{35 d^2}{12 (c+d x)^{3/2} (b c-a d)^3}+\frac{7 d}{4 (a+b x) (c+d x)^{3/2} (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.194914, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{35 b^{3/2} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{9/2}}+\frac{35 b d^2}{4 \sqrt{c+d x} (b c-a d)^4}+\frac{35 d^2}{12 (c+d x)^{3/2} (b c-a d)^3}+\frac{7 d}{4 (a+b x) (c+d x)^{3/2} (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^3*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 39.5389, size = 148, normalized size = 0.89 \[ \frac{35 b^{\frac{3}{2}} d^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 \left (a d - b c\right )^{\frac{9}{2}}} + \frac{35 b d^{2}}{4 \sqrt{c + d x} \left (a d - b c\right )^{4}} - \frac{35 d^{2}}{12 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{7 d}{4 \left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**3/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.474677, size = 143, normalized size = 0.86 \[ \frac{\sqrt{c+d x} \left (-\frac{6 b^2 (b c-a d)}{(a+b x)^2}+\frac{33 b^2 d}{a+b x}+\frac{8 d^2 (b c-a d)}{(c+d x)^2}+\frac{72 b d^2}{c+d x}\right )}{12 (b c-a d)^4}-\frac{35 b^{3/2} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^3*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.027, size = 206, normalized size = 1.2 \[ -{\frac{2\,{d}^{2}}{3\, \left ( ad-bc \right ) ^{3}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{{d}^{2}b}{ \left ( ad-bc \right ) ^{4}\sqrt{dx+c}}}+{\frac{11\,{d}^{2}{b}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{d}^{3}{b}^{2}a}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{13\,{d}^{2}{b}^{3}c}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{35\,{d}^{2}{b}^{2}}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^3/(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)^3*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236781, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^3*(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)**3/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.223968, size = 402, normalized size = 2.41 \[ \frac{35 \, b^{2} d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (9 \,{\left (d x + c\right )} b d^{2} + b c d^{2} - a d^{3}\right )}}{3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} + \frac{11 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} d^{2} - 13 \, \sqrt{d x + c} b^{3} c d^{2} + 13 \, \sqrt{d x + c} a b^{2} d^{3}}{4 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^3*(d*x + c)^(5/2)),x, algorithm="giac")
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