Optimal. Leaf size=197 \[ -\frac{A b-a B}{2 b (a+b x)^2 \sqrt{d+e x} (b d-a e)}-\frac{3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt{d+e x} (b d-a e)^2}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}} \]
[Out]
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Rubi [A] time = 0.404686, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{A b-a B}{2 b (a+b x)^2 \sqrt{d+e x} (b d-a e)}-\frac{3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt{d+e x} (b d-a e)^2}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^3*(d + e*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 42.7661, size = 182, normalized size = 0.92 \[ - \frac{3 e \left (5 A b e - B a e - 4 B b d\right )}{4 b \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{5 A b e - B a e - 4 B b d}{4 b \left (a + b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{A b - B a}{2 b \left (a + b x\right )^{2} \sqrt{d + e x} \left (a e - b d\right )} - \frac{3 e \left (5 A b e - B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \sqrt{b} \left (a e - b d\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**3/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.879222, size = 156, normalized size = 0.79 \[ \frac{1}{4} \left (\frac{\sqrt{d+e x} \left (\frac{2 (a B-A b) (b d-a e)}{(a+b x)^2}+\frac{-3 a B e+7 A b e-4 b B d}{a+b x}+\frac{8 e (A e-B d)}{d+e x}\right )}{(b d-a e)^3}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{7/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.03, size = 485, normalized size = 2.5 \[ -2\,{\frac{A{e}^{2}}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+2\,{\frac{eBd}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}-{\frac{7\,{b}^{2}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bba{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}Bde}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,Aab{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{9\,{b}^{2}Ad{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,B{a}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{Bbad{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{{b}^{2}eB{d}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,Ab{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{3\,Ba{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+3\,{\frac{bBde}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^3/(e*x+d)^(3/2),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)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234017, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)^(3/2)),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)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224927, size = 467, normalized size = 2.37 \[ -\frac{3 \,{\left (4 \, B b d e + B a e^{2} - 5 \, A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (B d e - A e^{2}\right )}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{x e + d}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} - 7 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} - \sqrt{x e + d} B a b d e^{2} + 9 \, \sqrt{x e + d} A b^{2} d e^{2} + 5 \, \sqrt{x e + d} B a^{2} e^{3} - 9 \, \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]