Optimal. Leaf size=123 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e}+\frac {a (d-e x)}{c \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1647, 844, 217, 206, 725} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e}+\frac {a (d-e x)}{c \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 844
Rule 1647
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\frac {a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {a^2 d e}{c d^2+a e^2}-a x}{(d+e x) \sqrt {a+c x^2}} \, dx}{a c}\\ &=\frac {a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {1}{\sqrt {a+c x^2}} \, dx}{c e}-\frac {d^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e \left (c d^2+a e^2\right )}\\ &=\frac {a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c e}+\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 \left (c d^2+a e^2\right )}\\ &=\frac {a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e}+\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 \left (c d^2+a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 153, normalized size = 1.24 \[ \frac {\frac {\sqrt {c} \left (a e (d-e x) \sqrt {a e^2+c d^2}+c d^3 \sqrt {a+c x^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )\right )}{\left (a e^2+c d^2\right )^{3/2}}+\sqrt {a} \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2} e \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 9.78, size = 1323, normalized size = 10.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 219, normalized size = 1.78 \[ -\frac {2 \, d^{3} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} e + a e^{3}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {\frac {{\left (a c^{2} d^{2} e^{3} + a^{2} c e^{5}\right )} x}{c^{4} d^{4} e^{2} + 2 \, a c^{3} d^{2} e^{4} + a^{2} c^{2} e^{6}} - \frac {a c^{2} d^{3} e^{2} + a^{2} c d e^{4}}{c^{4} d^{4} e^{2} + 2 \, a c^{3} d^{2} e^{4} + a^{2} c^{2} e^{6}}}{\sqrt {c x^{2} + a}} - \frac {e^{\left (-1\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 354, normalized size = 2.88 \[ -\frac {c \,d^{4} x}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a \,e^{3}}+\frac {d^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{2}}-\frac {d^{3}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{2}}+\frac {d^{2} x}{\sqrt {c \,x^{2}+a}\, a \,e^{3}}-\frac {x}{\sqrt {c \,x^{2}+a}\, c e}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}} e}+\frac {d}{\sqrt {c \,x^{2}+a}\, c \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 211, normalized size = 1.72 \[ -\frac {c d^{4} x}{\sqrt {c x^{2} + a} a c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{2} e^{5}} - \frac {d^{3}}{\sqrt {c x^{2} + a} c d^{2} e^{2} + \sqrt {c x^{2} + a} a e^{4}} + \frac {d^{2} x}{\sqrt {c x^{2} + a} a e^{3}} - \frac {x}{\sqrt {c x^{2} + a} c e} + \frac {\operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}} e} - \frac {d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{4}} + \frac {d}{\sqrt {c x^{2} + a} c e^{2}} \]
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 \[ \int \frac {x^3}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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