Optimal. Leaf size=98 \[ \frac {3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}}-\frac {3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {729, 723, 205} \begin {gather*} \frac {3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}}-\frac {3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 723
Rule 729
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx &=\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}+\frac {(3 d) \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac {3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac {\left (3 d \left (c d^2+a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c}\\ &=\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac {3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac {3 d \left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 127, normalized size = 1.30 \begin {gather*} \frac {\frac {\sqrt {a} \left (-2 a^3 e^3-a^2 c e \left (6 d^2+3 d e x+4 e^2 x^2\right )+a c^2 d x \left (5 d^2+3 e^2 x^2\right )+3 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt {c} d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 406, normalized size = 4.14 \begin {gather*} \left [-\frac {8 \, a^{3} c e^{3} x^{2} + 12 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 6 \, {\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} + 3 \, {\left (a^{2} c d^{3} + a^{3} d e^{2} + {\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \, {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{16 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac {4 \, a^{3} c e^{3} x^{2} + 6 \, a^{3} c d^{2} e + 2 \, a^{4} e^{3} - 3 \, {\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \, {\left (a^{2} c d^{3} + a^{3} d e^{2} + {\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \, {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{8 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 124, normalized size = 1.27 \begin {gather*} \frac {3 \, {\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} + \frac {3 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d x^{3} e^{2} + 5 \, a c^{2} d^{3} x - 4 \, a^{2} c x^{2} e^{3} - 3 \, a^{2} c d x e^{2} - 6 \, a^{2} c d^{2} e - 2 \, a^{3} e^{3}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 133, normalized size = 1.36 \begin {gather*} \frac {3 d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}+\frac {3 d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}+\frac {-\frac {e^{3} x^{2}}{2 c}+\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) d \,x^{3}}{8 a^{2}}-\frac {\left (3 a \,e^{2}-5 c \,d^{2}\right ) d x}{8 a c}-\frac {\left (a \,e^{2}+3 c \,d^{2}\right ) e}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 144, normalized size = 1.47 \begin {gather*} -\frac {4 \, a^{2} c e^{3} x^{2} + 6 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} - 3 \, {\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{3} - {\left (5 \, a c^{2} d^{3} - 3 \, a^{2} c d e^{2}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {3 \, {\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 125, normalized size = 1.28 \begin {gather*} \frac {3\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (c\,d^2+a\,e^2\right )}{8\,a^{5/2}\,c^{3/2}}-\frac {\frac {e^3\,x^2}{2\,c}+\frac {e\,\left (3\,c\,d^2+a\,e^2\right )}{4\,c^2}-\frac {3\,d\,x^3\,\left (c\,d^2+a\,e^2\right )}{8\,a^2}+\frac {d\,x\,\left (3\,a\,e^2-5\,c\,d^2\right )}{8\,a\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.27, size = 272, normalized size = 2.78 \begin {gather*} - \frac {3 d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (- \frac {3 a^{3} c d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac {3 d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (\frac {3 a^{3} c d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac {- 2 a^{3} e^{3} - 6 a^{2} c d^{2} e - 4 a^{2} c e^{3} x^{2} + x^{3} \left (3 a c^{2} d e^{2} + 3 c^{3} d^{3}\right ) + x \left (- 3 a^{2} c d e^{2} + 5 a c^{2} d^{3}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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