Optimal. Leaf size=88 \[ \frac {d e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {d-e x}{\sqrt {a+c x^2} \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {823, 12, 725, 206} \[ \frac {d e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {d-e x}{\sqrt {a+c x^2} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 823
Rubi steps
\begin {align*} \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=-\frac {d-e x}{\left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {\int \frac {a c d e}{(d+e x) \sqrt {a+c x^2}} \, dx}{a c \left (c d^2+a e^2\right )}\\ &=-\frac {d-e x}{\left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {(d e) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2}\\ &=-\frac {d-e x}{\left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {(d e) \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 )}{c d^2+a e^2}\\ &=-\frac {d-e x}{\left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {d e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 1.00 \[ \frac {e x-d}{\sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {d e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 425, normalized size = 4.83 \[ \left [\frac {{\left (c d e x^{2} + a d e\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (c d^{3} + a d e^{2} - {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} + {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{2}\right )}}, \frac {{\left (c d e x^{2} + a d e\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (c d^{3} + a d e^{2} - {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} + {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 162, normalized size = 1.84 \[ \frac {2 \, d \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e}{{\left (c d^{2} + a e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {\frac {{\left (c d^{2} e + a e^{3}\right )} x}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {c d^{3} + a d e^{2}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}}{\sqrt {c x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 283, normalized size = 3.22 \[ -\frac {c \,d^{2} x}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a e}+\frac {d \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {d}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {x}{\sqrt {c \,x^{2}+a}\, a e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 148, normalized size = 1.68 \[ -\frac {c d^{2} x}{\sqrt {c x^{2} + a} a c d^{2} e + \sqrt {c x^{2} + a} a^{2} e^{3}} - \frac {d}{\sqrt {c x^{2} + a} c d^{2} + \sqrt {c x^{2} + a} a e^{2}} + \frac {x}{\sqrt {c x^{2} + a} a e} - \frac {d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{2}} \]
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 \[ \int \frac {x}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {x}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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