Optimal. Leaf size=171 \[ \frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{2 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{5 c^3 d x}{e^6}+\frac{c^3 x^2}{2 e^5} \]
[Out]
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Rubi [A] time = 0.373612, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{2 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{5 c^3 d x}{e^6}+\frac{c^3 x^2}{2 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^3/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{5 c^{3} d x}{e^{6}} + \frac{c^{3} \int x\, dx}{e^{5}} + \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )} + \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{2 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{3}} - \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{2 e^{7} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{4 e^{7} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**3/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.12528, size = 185, normalized size = 1.08 \[ \frac{-a^3 e^6-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+12 c^2 (d+e x)^4 \left (a e^2+5 c d^2\right ) \log (d+e x)+a c^2 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )}{4 e^7 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^3/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.014, size = 268, normalized size = 1.6 \[{\frac{{c}^{3}{x}^{2}}{2\,{e}^{5}}}-5\,{\frac{{c}^{3}dx}{{e}^{6}}}-{\frac{3\,{a}^{2}c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-9\,{\frac{{d}^{2}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{4}{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{2}\ln \left ( ex+d \right ) a}{{e}^{5}}}+15\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}+2\,{\frac{{a}^{2}cd}{{e}^{3} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{d}^{3}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{3}}}+12\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}+20\,{\frac{{c}^{3}{d}^{3}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{a}^{3}}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{3\,{a}^{2}c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{d}^{4}a{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{3}{d}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^3/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.721937, size = 323, normalized size = 1.89 \[ \frac{57 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6} + 16 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} + 18 \, a c^{2} d^{2} e^{4} - a^{2} c e^{6}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 10 \, c^{3} d x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213975, size = 481, normalized size = 2.81 \[ \frac{2 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 68 \, c^{3} d^{2} e^{4} x^{4} + 57 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6} - 16 \,{\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} + 18 \, a c^{2} d^{2} e^{4} - a^{2} c e^{6}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.5111, size = 243, normalized size = 1.42 \[ - \frac{5 c^{3} d x}{e^{6}} + \frac{c^{3} x^{2}}{2 e^{5}} + \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{- a^{3} e^{6} - a^{2} c d^{2} e^{4} + 25 a c^{2} d^{4} e^{2} + 57 c^{3} d^{6} + x^{3} \left (48 a c^{2} d e^{5} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} + 108 a c^{2} d^{2} e^{4} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} c d e^{5} + 88 a c^{2} d^{3} e^{3} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**3/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.217883, size = 389, normalized size = 2.27 \[ \frac{1}{2} \,{\left (c^{3} - \frac{12 \, c^{3} d}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac{30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} + \frac{48 \, a c^{2} d e^{31}}{x e + d} - \frac{36 \, a c^{2} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{16 \, a c^{2} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{3 \, a c^{2} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{6 \, a^{2} c e^{33}}{{\left (x e + d\right )}^{2}} + \frac{8 \, a^{2} c d e^{33}}{{\left (x e + d\right )}^{3}} - \frac{3 \, a^{2} c d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac{a^{3} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^5,x, algorithm="giac")
[Out]