Optimal. Leaf size=278 \[ -\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.503308, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.68359, size = 166, normalized size = 0.6 \[ \frac{(a+b x)^5 \left (\frac{\sqrt{d+e x} \left (70 e^2 (a+b x)^2 (a e-b d)+56 e (a+b x) (b d-a e)^2-48 (b d-a e)^3+105 e^3 (a+b x)^3\right )}{3 (a+b x)^4 (b d-a e)^4}-\frac{35 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{9/2}}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 497, normalized size = 1.8 \[{\frac{bx+a}{192\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{4}{b}^{4}{e}^{4}+420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}a{b}^{3}{e}^{4}+105\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{3}{b}^{3}{e}^{3}+630\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+385\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{3}-70\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}{b}^{3}d{e}^{2}+420\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}+511\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{a}^{2}b{e}^{3}-252\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xa{b}^{2}d{e}^{2}+56\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{b}^{3}{d}^{2}e+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}+279\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}-326\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}+200\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e-48\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x)
[Out]
_______________________________________________________________________________________
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^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.22855, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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(1/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.240812, size = 757, normalized size = 2.72 \[ -\frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{105 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 385 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 385 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 1022 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 837 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 511 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 837 \, \sqrt{x e + d} a^{2} b d e^{6} + 279 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="giac")
[Out]