Optimal. Leaf size=111 \[ -\frac{2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac{2 B \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2}}-\frac{2 B \sqrt{d+e x}}{b^2 \sqrt{a+b x}} \]
[Out]
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Rubi [A] time = 0.166819, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac{2 B \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2}}-\frac{2 B \sqrt{d+e x}}{b^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[d + e*x])/(a + b*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 16.5122, size = 100, normalized size = 0.9 \[ - \frac{2 B \sqrt{d + e x}}{b^{2} \sqrt{a + b x}} + \frac{2 B \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{b^{\frac{5}{2}}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.189448, size = 128, normalized size = 1.15 \[ \frac{B \sqrt{e} \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{b^{5/2}}-\frac{2 \sqrt{d+e x} \left (B \left (-3 a^2 e+2 a b (d-2 e x)+3 b^2 d x\right )+A b^2 (d+e x)\right )}{3 b^2 (a+b x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[d + e*x])/(a + b*x)^(5/2),x]
[Out]
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Maple [B] time = 0.032, size = 503, normalized size = 4.5 \[{\frac{1}{ \left ( 3\,ae-3\,bd \right ){b}^{2}} \left ( 3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){x}^{2}a{b}^{2}{e}^{2}-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){x}^{2}{b}^{3}de+6\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{a}^{2}b{e}^{2}-6\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xa{b}^{2}de+2\,Ax{b}^{2}e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}{e}^{2}-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}bde-8\,Bxabe\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+6\,Bx{b}^{2}d\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+2\,A{b}^{2}d\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-6\,B{a}^{2}e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+4\,Babd\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) \sqrt{ex+d}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^(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)*sqrt(e*x + d)/(b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.573645, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a^{2} b d - B a^{3} e +{\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \,{\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt{\frac{e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{e}{b}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \,{\left (3 \, B a^{2} e -{\left (2 \, B a b + A b^{2}\right )} d -{\left (3 \, B b^{2} d -{\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{6 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e +{\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \,{\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}, \frac{3 \,{\left (B a^{2} b d - B a^{3} e +{\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \,{\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, b e x + b d + a e}{2 \, \sqrt{b x + a} \sqrt{e x + d} b \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (3 \, B a^{2} e -{\left (2 \, B a b + A b^{2}\right )} d -{\left (3 \, B b^{2} d -{\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e +{\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \,{\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(b*x + a)^(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)*(e*x+d)**(1/2)/(b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.578963, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(b*x + a)^(5/2),x, algorithm="giac")
[Out]