Optimal. Leaf size=90 \[ \frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {702, 635, 205, 260} \begin {gather*} \frac {e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 702
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{a+c x^2} \, dx &=\int \left (\frac {3 d e^2}{c}+\frac {e^3 x}{c}+\frac {c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c}+\frac {\int \frac {c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x}{a+c x^2} \, dx}{c}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c}+\frac {\left (d \left (c d^2-3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{c}+\frac {\left (e \left (3 c d^2-a e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{c}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c}+\frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.89 \begin {gather*} \frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (\left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )+c e x (6 d+e x)\right )}{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}{a+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 187, normalized size = 2.08 \begin {gather*} \left [\frac {a c e^{3} x^{2} + 6 \, a c d e^{2} x + {\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac {a c e^{3} x^{2} + 6 \, a c d e^{2} x + 2 \, {\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 78, normalized size = 0.87 \begin {gather*} \frac {{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {c x^{2} e^{3} + 6 \, c d x e^{2}}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 99, normalized size = 1.10 \begin {gather*} -\frac {3 a d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {e^{3} x^{2}}{2 c}+\frac {d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}-\frac {a \,e^{3} \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {3 d^{2} e \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {3 d \,e^{2} x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 79, normalized size = 0.88 \begin {gather*} \frac {{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {e^{3} x^{2} + 6 \, d e^{2} x}{2 \, c} + \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 91, normalized size = 1.01 \begin {gather*} \frac {e^3\,x^2}{2\,c}-\frac {\ln \left (c\,x^2+a\right )\,\left (4\,a^2\,c^2\,e^3-12\,a\,c^3\,d^2\,e\right )}{8\,a\,c^4}+\frac {3\,d\,e^2\,x}{c}-\frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,a\,e^2-c\,d^2\right )}{\sqrt {a}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 308, normalized size = 3.42 \begin {gather*} \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log {\left (x + \frac {- a^{2} e^{3} - 2 a c^{2} \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log {\left (x + \frac {- a^{2} e^{3} - 2 a c^{2} \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \frac {3 d e^{2} x}{c} + \frac {e^{3} x^{2}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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