Optimal. Leaf size=280 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5 (a+b x) (d+e x)^2}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}-\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+3 b B d)}{e^4 (a+b x)}+\frac{b^3 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)} \]
[Out]
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Rubi [A] time = 0.594779, antiderivative size = 280, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5 (a+b x) (d+e x)^2}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}-\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+3 b B d)}{e^4 (a+b x)}+\frac{b^3 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 52.5609, size = 270, normalized size = 0.96 \[ \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right )}{2 e^{3} \left (a e - b d\right )} + \frac{3 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right )}{e^{4}} + \frac{3 b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{5} \left (a + b x\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 e \left (d + e x\right )^{2} \left (a e - b d\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e + B a e - 2 B b d\right )}{e^{2} \left (d + e x\right ) \left (a e - b d\right )} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.279162, size = 256, normalized size = 0.91 \[ \frac{\sqrt{(a+b x)^2} \left (-a^3 e^3 (A e+B (d+2 e x))-3 a^2 b e^2 (A e (d+2 e x)-B d (3 d+4 e x))+3 a b^2 e \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )+6 b (d+e x)^2 (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)+b^3 \left (A e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+B \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )\right )\right )}{2 e^5 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^3,x]
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Maple [B] time = 0.027, size = 566, normalized size = 2. \[{\frac{-18\,B\ln \left ( ex+d \right ){x}^{2}a{b}^{2}d{e}^{3}+12\,A\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{3}+12\,B\ln \left ( ex+d \right ) x{a}^{2}bd{e}^{3}-36\,B\ln \left ( ex+d \right ) xa{b}^{2}{d}^{2}{e}^{2}-11\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+4\,A{x}^{2}{b}^{3}d{e}^{3}-Bd{e}^{3}{a}^{3}+9\,Aa{b}^{2}{d}^{2}{e}^{2}+B{x}^{4}{b}^{3}{e}^{4}+2\,A{x}^{3}{b}^{3}{e}^{4}+12\,B\ln \left ( ex+d \right ){b}^{3}{d}^{4}-2\,Bx{a}^{3}{e}^{4}+12\,B{x}^{2}a{b}^{2}d{e}^{3}-5\,A{b}^{3}{d}^{3}e+6\,B{x}^{3}a{b}^{2}{e}^{4}-4\,B{x}^{3}{b}^{3}d{e}^{3}+2\,Bx{b}^{3}{d}^{3}e+6\,A\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}{e}^{2}+12\,Axa{b}^{2}d{e}^{3}-A{a}^{3}{e}^{4}+7\,B{b}^{3}{d}^{4}-4\,Ax{b}^{3}{d}^{2}{e}^{2}+6\,B\ln \left ( ex+d \right ){a}^{2}b{d}^{2}{e}^{2}-18\,B\ln \left ( ex+d \right ) a{b}^{2}{d}^{3}e+12\,Bx{a}^{2}bd{e}^{3}-12\,Bxa{b}^{2}{d}^{2}{e}^{2}+9\,B{a}^{2}b{d}^{2}{e}^{2}-15\,Ba{b}^{2}{d}^{3}e+6\,A\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{4}-6\,A\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{3}+6\,B\ln \left ( ex+d \right ){x}^{2}{a}^{2}b{e}^{4}-12\,A\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}{e}^{2}+24\,B\ln \left ( ex+d \right ) x{b}^{3}{d}^{3}e+12\,B\ln \left ( ex+d \right ){x}^{2}{b}^{3}{d}^{2}{e}^{2}-6\,A\ln \left ( ex+d \right ){b}^{3}{d}^{3}e-3\,Ad{e}^{3}{a}^{2}b-6\,Ax{a}^{2}b{e}^{4}}{2\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
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)^3,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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.321963, size = 567, normalized size = 2.02 \[ \frac{B b^{3} e^{4} x^{4} + 7 \, B b^{3} d^{4} - A a^{3} e^{4} - 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 2 \,{\left (2 \, B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} -{\left (11 \, B b^{3} d^{2} e^{2} - 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3}\right )} x^{2} + 2 \,{\left (B b^{3} d^{3} e - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 6 \,{\left (2 \, B b^{3} d^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e +{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} +{\left (2 \, B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} +{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 2 \,{\left (2 \, B b^{3} d^{3} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} +{\left (B a^{2} b + A a b^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
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)^3,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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.288759, size = 563, normalized size = 2.01 \[ 3 \,{\left (2 \, B b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 3 \, B a b^{2} d e{\rm sign}\left (b x + a\right ) - A b^{3} d e{\rm sign}\left (b x + a\right ) + B a^{2} b e^{2}{\rm sign}\left (b x + a\right ) + A a b^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) - 6 \, B b^{3} d x e^{2}{\rm sign}\left (b x + a\right ) + 6 \, B a b^{2} x e^{3}{\rm sign}\left (b x + a\right ) + 2 \, A b^{3} x e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )} + \frac{{\left (7 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) - 15 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) - 5 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 9 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 9 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) - 3 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) - A a^{3} e^{4}{\rm sign}\left (b x + a\right ) + 2 \,{\left (4 \, B b^{3} d^{3} e{\rm sign}\left (b x + a\right ) - 9 \, B a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 3 \, A b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, B a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 6 \, A a b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - B a^{3} e^{4}{\rm sign}\left (b x + a\right ) - 3 \, A a^{2} b e^{4}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
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)^3,x, algorithm="giac")
[Out]