Optimal. Leaf size=284 \[ -\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{2 e^5 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x) (d+e x)^3}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x) (-3 a B e-A b e+4 b B d)}{e^5 (a+b x)}+\frac{b^3 B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)} \]
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Rubi [A] time = 0.609263, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{2 e^5 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x) (d+e x)^3}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x) (-3 a B e-A b e+4 b B d)}{e^5 (a+b x)}+\frac{b^3 B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 54.8286, size = 286, normalized size = 1.01 \[ \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + 3 B a e - 4 B b d\right )}{e^{4} \left (a e - b d\right )} + \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + 3 B a e - 4 B b d\right ) \log{\left (d + e x \right )}}{e^{5} \left (a + b x\right )} - \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + 3 B a e - 4 B b d\right )}{6 e^{3} \left (d + e x\right ) \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 e \left (d + e x\right )^{3} \left (a e - b d\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{6 e^{2} \left (d + e x\right )^{2} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.279105, size = 251, normalized size = 0.88 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 e^3 (2 A e+B (d+3 e x))+3 a^2 b e^2 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+3 a b^2 e \left (2 A e \left (d^2+3 d e x+3 e^2 x^2\right )-B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )+6 b^2 (d+e x)^3 \log (d+e x) (-3 a B e-A b e+4 b B d)+b^3 \left (2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )-A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )\right )}{6 e^5 (a+b x) (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.027, size = 512, normalized size = 1.8 \[{\frac{54\,B\ln \left ( ex+d \right ){x}^{2}a{b}^{2}d{e}^{3}+54\,B\ln \left ( ex+d \right ) xa{b}^{2}{d}^{2}{e}^{2}-18\,B{x}^{2}{a}^{2}b{e}^{4}-18\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+18\,A{x}^{2}{b}^{3}d{e}^{3}-Bd{e}^{3}{a}^{3}-6\,Aa{b}^{2}{d}^{2}{e}^{2}+6\,B{x}^{4}{b}^{3}{e}^{4}-24\,B\ln \left ( ex+d \right ){b}^{3}{d}^{4}-3\,Bx{a}^{3}{e}^{4}+54\,B{x}^{2}a{b}^{2}d{e}^{3}+11\,A{b}^{3}{d}^{3}e+18\,B{x}^{3}{b}^{3}d{e}^{3}-54\,Bx{b}^{3}{d}^{3}e-18\,Axa{b}^{2}d{e}^{3}-2\,A{a}^{3}{e}^{4}-26\,B{b}^{3}{d}^{4}+27\,Ax{b}^{3}{d}^{2}{e}^{2}+18\,B\ln \left ( ex+d \right ) a{b}^{2}{d}^{3}e-18\,Bx{a}^{2}bd{e}^{3}+81\,Bxa{b}^{2}{d}^{2}{e}^{2}-6\,B{a}^{2}b{d}^{2}{e}^{2}+33\,Ba{b}^{2}{d}^{3}e+18\,A\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{3}+18\,A\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}{e}^{2}-72\,B\ln \left ( ex+d \right ) x{b}^{3}{d}^{3}e-72\,B\ln \left ( ex+d \right ){x}^{2}{b}^{3}{d}^{2}{e}^{2}+18\,B\ln \left ( ex+d \right ){x}^{3}a{b}^{2}{e}^{4}-24\,B\ln \left ( ex+d \right ){x}^{3}{b}^{3}d{e}^{3}+6\,A\ln \left ( ex+d \right ){b}^{3}{d}^{3}e-18\,A{x}^{2}a{b}^{2}{e}^{4}-3\,Ad{e}^{3}{a}^{2}b+6\,A\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{4}-9\,Ax{a}^{2}b{e}^{4}}{6\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^4,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^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278164, size = 548, normalized size = 1.93 \[ \frac{6 \, B b^{3} e^{4} x^{4} + 18 \, B b^{3} d e^{3} x^{3} - 26 \, B b^{3} d^{4} - 2 \, A a^{3} e^{4} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 6 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 18 \,{\left (B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} +{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 3 \,{\left (18 \, B b^{3} d^{3} e - 9 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x - 6 \,{\left (4 \, B b^{3} d^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e +{\left (4 \, B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (4 \, B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3}\right )} x^{2} + 3 \,{\left (4 \, B b^{3} d^{3} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^4,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**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.285886, size = 555, normalized size = 1.95 \[ B b^{3} x e^{\left (-4\right )}{\rm sign}\left (b x + a\right ) -{\left (4 \, B b^{3} d{\rm sign}\left (b x + a\right ) - 3 \, B a b^{2} e{\rm sign}\left (b x + a\right ) - A b^{3} e{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (26 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) - 33 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) - 11 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right ) + 18 \,{\left (2 \, B b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 3 \, B a b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - A b^{3} d e^{3}{\rm sign}\left (b x + a\right ) + B a^{2} b e^{4}{\rm sign}\left (b x + a\right ) + A a b^{2} e^{4}{\rm sign}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (20 \, B b^{3} d^{3} e{\rm sign}\left (b x + a\right ) - 27 \, B a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 9 \, A b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, B a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 6 \, A a b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + B a^{3} e^{4}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b e^{4}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-5\right )}}{6 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^4,x, algorithm="giac")
[Out]