Optimal. Leaf size=240 \[ -\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^4}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]
[Out]
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Rubi [A] time = 0.51879, antiderivative size = 240, 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 \[ -\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^4}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 54.8479, size = 230, normalized size = 0.96 \[ \frac{5 \sqrt{b} e \left (7 A b e - 3 B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \left (a e - b d\right )^{\frac{9}{2}}} + \frac{5 e \left (7 A b e - 3 B a e - 4 B b d\right )}{4 \sqrt{d + e x} \left (a e - b d\right )^{4}} - \frac{5 e \left (7 A b e - 3 B a e - 4 B b d\right )}{12 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} + \frac{7 A b e - 3 B a e - 4 B b d}{4 b \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{2 b \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**3/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 1.36517, size = 195, normalized size = 0.81 \[ \frac{\sqrt{d+e x} \left (-\frac{6 b (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{8 e (b d-a e) (A e-B d)}{(d+e x)^2}-\frac{3 b (7 a B e-11 A b e+4 b B d)}{a+b x}+\frac{24 e (-a B e+3 A b e-2 b B d)}{d+e x}\right )}{12 (b d-a e)^4}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x]
[Out]
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Maple [B] time = 0.037, size = 568, normalized size = 2.4 \[ -{\frac{2\,A{e}^{2}}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,eBd}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{Ab{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-2\,{\frac{Ba{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-4\,{\frac{bBde}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+{\frac{11\,{b}^{3}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}Ba{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{3}Bde}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{b}^{2}Aa{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{13\,{b}^{3}Ad{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{9\,bB{a}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,{b}^{2}Bad{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{{b}^{3}B{d}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{b}^{2}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{15\,Bba{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-5\,{\frac{{b}^{2}Bde}{ \left ( ae-bd \right ) ^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^3/(e*x+d)^(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((B*x + A)/((b*x + a)^3*(e*x + d)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240887, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)^(5/2)),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((B*x+A)/(b*x+a)**3/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227213, size = 606, normalized size = 2.52 \[ -\frac{5 \,{\left (4 \, B b^{2} d e + 3 \, B a b e^{2} - 7 \, A b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} B b d e + B b d^{2} e + 3 \,{\left (x e + d\right )} B a e^{2} - 9 \,{\left (x e + d\right )} A b e^{2} - B a d e^{2} - A b d e^{2} + A a e^{3}\right )}}{3 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d e - 4 \, \sqrt{x e + d} B b^{3} d^{2} e + 7 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} e^{2} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} e^{2} - 5 \, \sqrt{x e + d} B a b^{2} d e^{2} + 13 \, \sqrt{x e + d} A b^{3} d e^{2} + 9 \, \sqrt{x e + d} B a^{2} b e^{3} - 13 \, \sqrt{x e + d} A a b^{2} e^{3}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)^(5/2)),x, algorithm="giac")
[Out]