Optimal. Leaf size=109 \[ -\frac {d e^2 x}{2 a c}-\frac {(a e-c d x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac {d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}+\frac {e^3 \log \left (a+c x^2\right )}{2 c^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {753, 788, 649,
211, 266} \begin {gather*} \frac {d \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 a e^2+c d^2\right )}{2 a^{3/2} c^{3/2}}+\frac {e^3 \log \left (a+c x^2\right )}{2 c^2}-\frac {d e^2 x}{2 a c}-\frac {(d+e x)^2 (a e-c d x)}{2 a c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 753
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {(d+e x) \left (c d^2+2 a e^2-c d e x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {d e^2 x}{2 a c}-\frac {(a e-c d x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {a c d e^2+c d \left (c d^2+2 a e^2\right )+c \left (-c d^2 e+e \left (c d^2+2 a e^2\right )\right ) x}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac {d e^2 x}{2 a c}-\frac {(a e-c d x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac {e^3 \int \frac {x}{a+c x^2} \, dx}{c}+\frac {\left (d \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {d e^2 x}{2 a c}-\frac {(a e-c d x) (d+e x)^2}{2 a c \left (a+c x^2\right )}+\frac {d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}+\frac {e^3 \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 107, normalized size = 0.98 \begin {gather*} \frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+\frac {\sqrt {a} \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)+a e^3 \left (a+c x^2\right ) \log \left (a+c x^2\right )\right )}{a+c x^2}}{2 a^{3/2} c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 107, normalized size = 0.98
method | result | size |
default | \(\frac {-\frac {d \left (3 e^{2} a -c \,d^{2}\right ) x}{2 a c}+\frac {e \left (e^{2} a -3 c \,d^{2}\right )}{2 c^{2}}}{c \,x^{2}+a}+\frac {\frac {a \,e^{3} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (3 a d \,e^{2}+c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a c}\) | \(107\) |
risch | \(\frac {-\frac {d \left (3 e^{2} a -c \,d^{2}\right ) x}{2 a c}+\frac {e \left (e^{2} a -3 c \,d^{2}\right )}{2 c^{2}}}{c \,x^{2}+a}+\frac {\ln \left (3 a^{2} d \,e^{2}+d^{3} a c -\sqrt {-a c \,d^{2} \left (3 e^{2} a +c \,d^{2}\right )^{2}}\, x \right ) e^{3}}{2 c^{2}}+\frac {\ln \left (3 a^{2} d \,e^{2}+d^{3} a c -\sqrt {-a c \,d^{2} \left (3 e^{2} a +c \,d^{2}\right )^{2}}\, x \right ) \sqrt {-a c \,d^{2} \left (3 e^{2} a +c \,d^{2}\right )^{2}}}{4 a^{2} c^{2}}+\frac {\ln \left (3 a^{2} d \,e^{2}+d^{3} a c +\sqrt {-a c \,d^{2} \left (3 e^{2} a +c \,d^{2}\right )^{2}}\, x \right ) e^{3}}{2 c^{2}}-\frac {\ln \left (3 a^{2} d \,e^{2}+d^{3} a c +\sqrt {-a c \,d^{2} \left (3 e^{2} a +c \,d^{2}\right )^{2}}\, x \right ) \sqrt {-a c \,d^{2} \left (3 e^{2} a +c \,d^{2}\right )^{2}}}{4 a^{2} c^{2}}\) | \(302\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 105, normalized size = 0.96 \begin {gather*} -\frac {3 \, a c d^{2} e - a^{2} e^{3} - {\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {{\left (c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.50, size = 290, normalized size = 2.66 \begin {gather*} \left [\frac {2 \, a c^{2} d^{3} x - 6 \, a^{2} c d x e^{2} - 6 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} + 2 \, {\left (a^{2} c x^{2} + a^{3}\right )} e^{3} \log \left (c x^{2} + a\right ) - {\left (c^{2} d^{3} x^{2} + a c d^{3} + 3 \, {\left (a c d x^{2} + a^{2} d\right )} e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, \frac {a c^{2} d^{3} x - 3 \, a^{2} c d x e^{2} - 3 \, a^{2} c d^{2} e + a^{3} e^{3} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{3} \log \left (c x^{2} + a\right ) + {\left (c^{2} d^{3} x^{2} + a c d^{3} + 3 \, {\left (a c d x^{2} + a^{2} d\right )} e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 298 vs.
\(2 (95) = 190\).
time = 0.53, size = 298, normalized size = 2.73 \begin {gather*} \left (\frac {e^{3}}{2 c^{2}} - \frac {d \sqrt {- a^{3} c^{5}} \cdot \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) \log {\left (x + \frac {4 a^{2} c^{2} \left (\frac {e^{3}}{2 c^{2}} - \frac {d \sqrt {- a^{3} c^{5}} \cdot \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) - 2 a^{2} e^{3}}{3 a c d e^{2} + c^{2} d^{3}} \right )} + \left (\frac {e^{3}}{2 c^{2}} + \frac {d \sqrt {- a^{3} c^{5}} \cdot \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) \log {\left (x + \frac {4 a^{2} c^{2} \left (\frac {e^{3}}{2 c^{2}} + \frac {d \sqrt {- a^{3} c^{5}} \cdot \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) - 2 a^{2} e^{3}}{3 a c d e^{2} + c^{2} d^{3}} \right )} + \frac {a^{2} e^{3} - 3 a c d^{2} e + x \left (- 3 a c d e^{2} + c^{2} d^{3}\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 104, normalized size = 0.95 \begin {gather*} \frac {e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {{\left (c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} + \frac {{\left (c d^{3} - 3 \, a d e^{2}\right )} x - \frac {3 \, a c d^{2} e - a^{2} e^{3}}{c}}{2 \, {\left (c x^{2} + a\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 143, normalized size = 1.31 \begin {gather*} \frac {d^3\,x}{2\,\left (a^2+c\,a\,x^2\right )}-\frac {3\,d^2\,e}{2\,\left (c^2\,x^2+a\,c\right )}+\frac {e^3\,\ln \left (c\,x^2+a\right )}{2\,c^2}+\frac {a\,e^3}{2\,\left (c^3\,x^2+a\,c^2\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {c}}-\frac {3\,d\,e^2\,x}{2\,\left (c^2\,x^2+a\,c\right )}+\frac {3\,d\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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