Optimal. Leaf size=309 \[ -\frac{c x \left (6 B d \left (a e^2+c d^2\right )^2-A e \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}-\frac{c x^2 \left (2 A c d e \left (3 a e^2+2 c d^2\right )-B \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{2 e^6}-\frac{c^2 x^4 \left (2 A c d e-3 B \left (a e^2+c d^2\right )\right )}{4 e^4}+\frac{c^2 x^3 \left (3 A e \left (a e^2+c d^2\right )-B \left (6 a d e^2+4 c d^3\right )\right )}{3 e^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 (d+e x)}+\frac{\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}-\frac{c^3 x^5 (2 B d-A e)}{5 e^3}+\frac{B c^3 x^6}{6 e^2} \]
[Out]
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Rubi [A] time = 0.975074, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{c x \left (6 B d \left (a e^2+c d^2\right )^2-A e \left (3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right )\right )}{e^7}+\frac{c x^2 \left (3 a^2 B e^4-6 a A c d e^3+9 a B c d^2 e^2-4 A c^2 d^3 e+5 B c^2 d^4\right )}{2 e^6}-\frac{c^2 x^4 \left (2 A c d e-3 B \left (a e^2+c d^2\right )\right )}{4 e^4}-\frac{c^2 x^3 \left (-3 a A e^3+6 a B d e^2-3 A c d^2 e+4 B c d^3\right )}{3 e^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 (d+e x)}+\frac{\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}-\frac{c^3 x^5 (2 B d-A e)}{5 e^3}+\frac{B c^3 x^6}{6 e^2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} x^{6}}{6 e^{2}} + \frac{c^{3} x^{5} \left (A e - 2 B d\right )}{5 e^{3}} + \frac{c^{2} x^{4} \left (- 2 A c d e + 3 B a e^{2} + 3 B c d^{2}\right )}{4 e^{4}} + \frac{c^{2} x^{3} \left (3 A a e^{3} + 3 A c d^{2} e - 6 B a d e^{2} - 4 B c d^{3}\right )}{3 e^{5}} + \frac{c \left (- 6 A a c d e^{3} - 4 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 9 B a c d^{2} e^{2} + 5 B c^{2} d^{4}\right ) \int x\, dx}{e^{6}} + \frac{\left (3 A a^{2} e^{5} + 9 A a c d^{2} e^{3} + 5 A c^{2} d^{4} e - 6 B a^{2} d e^{4} - 12 B a c d^{3} e^{2} - 6 B c^{2} d^{5}\right ) \int c\, dx}{e^{7}} + \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{8}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{e^{8} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.547581, size = 405, normalized size = 1.31 \[ \frac{6 A e \left (-10 a^3 e^6+30 a^2 c e^4 \left (-d^2+d e x+e^2 x^2\right )+10 a c^2 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+c^3 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )\right )+B \left (60 a^3 d e^6+90 a^2 c e^4 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+15 a c^2 e^2 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+c^3 \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )+60 (d+e x) \left (a e^2+c d^2\right )^2 \log (d+e x) \left (a B e^2-6 A c d e+7 B c d^2\right )}{60 e^8 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^2,x]
[Out]
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Maple [A] time = 0.018, size = 558, normalized size = 1.8 \[{\frac{aA{c}^{2}{x}^{3}}{{e}^{2}}}+{\frac{B{c}^{3}{x}^{6}}{6\,{e}^{2}}}-12\,{\frac{\ln \left ( ex+d \right ) Aa{c}^{2}{d}^{3}}{{e}^{5}}}+9\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}c{d}^{2}}{{e}^{4}}}-{\frac{A{a}^{3}}{e \left ( ex+d \right ) }}+{\frac{\ln \left ( ex+d \right ) B{a}^{3}}{{e}^{2}}}-{\frac{4\,B{c}^{3}{x}^{3}{d}^{3}}{3\,{e}^{5}}}+{\frac{A{c}^{3}{x}^{3}{d}^{2}}{{e}^{4}}}-6\,{\frac{{d}^{5}\ln \left ( ex+d \right ) A{c}^{3}}{{e}^{7}}}+7\,{\frac{{d}^{6}\ln \left ( ex+d \right ) B{c}^{3}}{{e}^{8}}}-{\frac{{d}^{6}A{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{B{c}^{3}{d}^{7}}{{e}^{8} \left ( ex+d \right ) }}+3\,{\frac{Ba{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+3\,{\frac{B{a}^{2}c{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-2\,{\frac{aB{c}^{2}{x}^{3}d}{{e}^{3}}}-3\,{\frac{A{d}^{4}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{A{c}^{3}{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,{a}^{2}Bc{x}^{2}}{2\,{e}^{2}}}+3\,{\frac{{a}^{2}Acx}{{e}^{2}}}+{\frac{9\,B{c}^{2}{x}^{2}a{d}^{2}}{2\,{e}^{4}}}-6\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}cd}{{e}^{3}}}-{\frac{A{c}^{3}{x}^{4}d}{2\,{e}^{3}}}+{\frac{3\,B{c}^{3}{x}^{4}{d}^{2}}{4\,{e}^{4}}}-6\,{\frac{B{c}^{3}{d}^{5}x}{{e}^{7}}}-2\,{\frac{A{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}+{\frac{5\,B{c}^{3}{x}^{2}{d}^{4}}{2\,{e}^{6}}}-{\frac{2\,B{c}^{3}{x}^{5}d}{5\,{e}^{3}}}+{\frac{Bd{a}^{3}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{3\,aB{c}^{2}{x}^{4}}{4\,{e}^{2}}}-12\,{\frac{Ba{c}^{2}{d}^{3}x}{{e}^{5}}}+15\,{\frac{\ln \left ( ex+d \right ) Ba{c}^{2}{d}^{4}}{{e}^{6}}}-3\,{\frac{A{d}^{2}{a}^{2}c}{{e}^{3} \left ( ex+d \right ) }}+5\,{\frac{A{d}^{4}{c}^{3}x}{{e}^{6}}}-3\,{\frac{aA{c}^{2}{x}^{2}d}{{e}^{3}}}+9\,{\frac{A{d}^{2}a{c}^{2}x}{{e}^{4}}}-6\,{\frac{B{a}^{2}cdx}{{e}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^3/(e*x+d)^2,x)
[Out]
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Maxima [A] time = 0.703578, size = 616, normalized size = 1.99 \[ \frac{B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}}{e^{9} x + d e^{8}} + \frac{10 \, B c^{3} e^{5} x^{6} - 12 \,{\left (2 \, B c^{3} d e^{4} - A c^{3} e^{5}\right )} x^{5} + 15 \,{\left (3 \, B c^{3} d^{2} e^{3} - 2 \, A c^{3} d e^{4} + 3 \, B a c^{2} e^{5}\right )} x^{4} - 20 \,{\left (4 \, B c^{3} d^{3} e^{2} - 3 \, A c^{3} d^{2} e^{3} + 6 \, B a c^{2} d e^{4} - 3 \, A a c^{2} e^{5}\right )} x^{3} + 30 \,{\left (5 \, B c^{3} d^{4} e - 4 \, A c^{3} d^{3} e^{2} + 9 \, B a c^{2} d^{2} e^{3} - 6 \, A a c^{2} d e^{4} + 3 \, B a^{2} c e^{5}\right )} x^{2} - 60 \,{\left (6 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 12 \, B a c^{2} d^{3} e^{2} - 9 \, A a c^{2} d^{2} e^{3} + 6 \, B a^{2} c d e^{4} - 3 \, A a^{2} c e^{5}\right )} x}{60 \, e^{7}} + \frac{{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284037, size = 838, normalized size = 2.71 \[ \frac{10 \, B c^{3} e^{7} x^{7} + 60 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e + 180 \, B a c^{2} d^{5} e^{2} - 180 \, A a c^{2} d^{4} e^{3} + 180 \, B a^{2} c d^{3} e^{4} - 180 \, A a^{2} c d^{2} e^{5} + 60 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7} - 2 \,{\left (7 \, B c^{3} d e^{6} - 6 \, A c^{3} e^{7}\right )} x^{6} + 3 \,{\left (7 \, B c^{3} d^{2} e^{5} - 6 \, A c^{3} d e^{6} + 15 \, B a c^{2} e^{7}\right )} x^{5} - 5 \,{\left (7 \, B c^{3} d^{3} e^{4} - 6 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 12 \, A a c^{2} e^{7}\right )} x^{4} + 10 \,{\left (7 \, B c^{3} d^{4} e^{3} - 6 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} - 30 \,{\left (7 \, B c^{3} d^{5} e^{2} - 6 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} - 6 \, A a^{2} c e^{7}\right )} x^{2} - 60 \,{\left (6 \, B c^{3} d^{6} e - 5 \, A c^{3} d^{5} e^{2} + 12 \, B a c^{2} d^{4} e^{3} - 9 \, A a c^{2} d^{3} e^{4} + 6 \, B a^{2} c d^{2} e^{5} - 3 \, A a^{2} c d e^{6}\right )} x + 60 \,{\left (7 \, B c^{3} d^{7} - 6 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} - 6 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} +{\left (7 \, B c^{3} d^{6} e - 6 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x\right )} \log \left (e x + d\right )}{60 \,{\left (e^{9} x + d e^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.19119, size = 442, normalized size = 1.43 \[ \frac{B c^{3} x^{6}}{6 e^{2}} + \frac{- A a^{3} e^{7} - 3 A a^{2} c d^{2} e^{5} - 3 A a c^{2} d^{4} e^{3} - A c^{3} d^{6} e + B a^{3} d e^{6} + 3 B a^{2} c d^{3} e^{4} + 3 B a c^{2} d^{5} e^{2} + B c^{3} d^{7}}{d e^{8} + e^{9} x} - \frac{x^{5} \left (- A c^{3} e + 2 B c^{3} d\right )}{5 e^{3}} + \frac{x^{4} \left (- 2 A c^{3} d e + 3 B a c^{2} e^{2} + 3 B c^{3} d^{2}\right )}{4 e^{4}} - \frac{x^{3} \left (- 3 A a c^{2} e^{3} - 3 A c^{3} d^{2} e + 6 B a c^{2} d e^{2} + 4 B c^{3} d^{3}\right )}{3 e^{5}} + \frac{x^{2} \left (- 6 A a c^{2} d e^{3} - 4 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 9 B a c^{2} d^{2} e^{2} + 5 B c^{3} d^{4}\right )}{2 e^{6}} - \frac{x \left (- 3 A a^{2} c e^{5} - 9 A a c^{2} d^{2} e^{3} - 5 A c^{3} d^{4} e + 6 B a^{2} c d e^{4} + 12 B a c^{2} d^{3} e^{2} + 6 B c^{3} d^{5}\right )}{e^{7}} + \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.284006, size = 728, normalized size = 2.36 \[ \frac{1}{60} \,{\left (10 \, B c^{3} - \frac{12 \,{\left (7 \, B c^{3} d e - A c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{45 \,{\left (7 \, B c^{3} d^{2} e^{2} - 2 \, A c^{3} d e^{3} + B a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{20 \,{\left (35 \, B c^{3} d^{3} e^{3} - 15 \, A c^{3} d^{2} e^{4} + 15 \, B a c^{2} d e^{5} - 3 \, A a c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{30 \,{\left (35 \, B c^{3} d^{4} e^{4} - 20 \, A c^{3} d^{3} e^{5} + 30 \, B a c^{2} d^{2} e^{6} - 12 \, A a c^{2} d e^{7} + 3 \, B a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac{180 \,{\left (7 \, B c^{3} d^{5} e^{5} - 5 \, A c^{3} d^{4} e^{6} + 10 \, B a c^{2} d^{3} e^{7} - 6 \, A a c^{2} d^{2} e^{8} + 3 \, B a^{2} c d e^{9} - A a^{2} c e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}\right )}{\left (x e + d\right )}^{6} e^{\left (-8\right )} -{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} e^{\left (-8\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B c^{3} d^{7} e^{6}}{x e + d} - \frac{A c^{3} d^{6} e^{7}}{x e + d} + \frac{3 \, B a c^{2} d^{5} e^{8}}{x e + d} - \frac{3 \, A a c^{2} d^{4} e^{9}}{x e + d} + \frac{3 \, B a^{2} c d^{3} e^{10}}{x e + d} - \frac{3 \, A a^{2} c d^{2} e^{11}}{x e + d} + \frac{B a^{3} d e^{12}}{x e + d} - \frac{A a^{3} e^{13}}{x e + d}\right )} e^{\left (-14\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^2,x, algorithm="giac")
[Out]