Optimal. Leaf size=152 \[ \frac {\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac {d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3 \sqrt {a e^2+c d^2}}-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2} \]
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Rubi [A] time = 0.27, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1654, 844, 217, 206, 725} \[ \frac {\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac {d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3 \sqrt {a e^2+c d^2}}-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 844
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx &=\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\int \frac {-a d e^2-e \left (c d^2+a e^2\right ) x-3 c d e^2 x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c e^3}\\ &=-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\int \frac {-a c d e^4+c e^3 \left (2 c d^2-a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c^2 e^5}\\ &=-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}-\frac {d^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^3}+\frac {\left (2 c d^2-a e^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c e^3}\\ &=-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^3}+\frac {\left (2 c d^2-a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c e^3}\\ &=-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac {d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \sqrt {c d^2+a e^2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 131, normalized size = 0.86 \[ \frac {\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\sqrt {c} \left (\frac {2 c d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\sqrt {a e^2+c d^2}}+e \sqrt {a+c x^2} (e x-2 d)\right )}{2 c^{3/2} e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 5.57, size = 924, normalized size = 6.08 \[ \left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {\sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 129, normalized size = 0.85 \[ -\frac {2 \, d^{3} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-3\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{2} \, \sqrt {c x^{2} + a} {\left (\frac {x e^{\left (-1\right )}}{c} - \frac {2 \, d e^{\left (-2\right )}}{c}\right )} - \frac {{\left (2 \, c d^{2} - a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 217, normalized size = 1.43 \[ \frac {d^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{4}}-\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}} e}+\frac {d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}\, e^{3}}+\frac {\sqrt {c \,x^{2}+a}\, x}{2 c e}-\frac {\sqrt {c \,x^{2}+a}\, d}{c \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 130, normalized size = 0.86 \[ \frac {\sqrt {c x^{2} + a} x}{2 \, c e} + \frac {d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c} e^{3}} - \frac {a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {3}{2}} e} - \frac {d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{\sqrt {a + \frac {c d^{2}}{e^{2}}} e^{4}} - \frac {\sqrt {c x^{2} + a} d}{c e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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