Optimal. Leaf size=172 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]
[Out]
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Rubi [A] time = 0.376933, antiderivative size = 172, 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 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^2}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{3 c d \left (a e^2+c d^2\right )^2}{2 e^7 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^3/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{6 c^{3} d \log{\left (d + e x \right )}}{e^{7}} + \frac{2 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )^{2}} - \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )} + \frac{3 c d \left (a e^{2} + c d^{2}\right )^{2}}{2 e^{7} \left (d + e x\right )^{4}} - \frac{c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )^{3}} + \frac{\int c^{3}\, dx}{e^{6}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**3/(e*x+d)**6,x)
[Out]
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Mathematica [A] time = 0.176314, size = 182, normalized size = 1.06 \[ -\frac{2 a^3 e^6+a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+60 c^3 d (d+e x)^5 \log (d+e x)}{10 e^7 (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^3/(d + e*x)^6,x]
[Out]
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Maple [A] time = 0.016, size = 272, normalized size = 1.6 \[{\frac{{c}^{3}x}{{e}^{6}}}-{\frac{{a}^{3}}{5\,e \left ( ex+d \right ) ^{5}}}-{\frac{3\,{a}^{2}c{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{d}^{4}a{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}{d}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+6\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{c}^{3}{d}^{3}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-6\,{\frac{{c}^{3}d\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{a}^{2}c}{{e}^{3} \left ( ex+d \right ) ^{3}}}-6\,{\frac{{d}^{2}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{d}^{4}{c}^{3}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-3\,{\frac{a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{d}^{2}{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{3\,{a}^{2}cd}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+3\,{\frac{{d}^{3}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{c}^{3}{d}^{5}}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^3/(e*x+d)^6,x)
[Out]
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Maxima [A] time = 0.722311, size = 332, normalized size = 1.93 \[ -\frac{87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 30 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \,{\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 10 \,{\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac{c^{3} x}{e^{6}} - \frac{6 \, c^{3} d \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212685, size = 440, normalized size = 2.56 \[ \frac{10 \, c^{3} e^{6} x^{6} + 50 \, c^{3} d e^{5} x^{5} - 87 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 10 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \,{\left (20 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} - 10 \,{\left (60 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} - 5 \,{\left (75 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x - 60 \,{\left (c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{2} e^{4} x^{4} + 10 \, c^{3} d^{3} e^{3} x^{3} + 10 \, c^{3} d^{4} e^{2} x^{2} + 5 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 19.6733, size = 257, normalized size = 1.49 \[ - \frac{6 c^{3} d \log{\left (d + e x \right )}}{e^{7}} + \frac{c^{3} x}{e^{6}} - \frac{2 a^{3} e^{6} + a^{2} c d^{2} e^{4} + 6 a c^{2} d^{4} e^{2} + 87 c^{3} d^{6} + x^{4} \left (30 a c^{2} e^{6} + 150 c^{3} d^{2} e^{4}\right ) + x^{3} \left (60 a c^{2} d e^{5} + 500 c^{3} d^{3} e^{3}\right ) + x^{2} \left (10 a^{2} c e^{6} + 60 a c^{2} d^{2} e^{4} + 650 c^{3} d^{4} e^{2}\right ) + x \left (5 a^{2} c d e^{5} + 30 a c^{2} d^{3} e^{3} + 385 c^{3} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**3/(e*x+d)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.212108, size = 254, normalized size = 1.48 \[ -6 \, c^{3} d e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + c^{3} x e^{\left (-6\right )} - \frac{{\left (87 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 30 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 20 \,{\left (25 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 2 \, a^{3} e^{6} + 10 \,{\left (65 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (77 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3/(e*x + d)^6,x, algorithm="giac")
[Out]