Optimal. Leaf size=137 \[ -\frac {d^2 \sqrt {a+c x^2}}{e (d+e x) \left (a e^2+c d^2\right )}+\frac {d \left (2 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2} \]
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Rubi [A] time = 0.17, antiderivative size = 137, 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 = {1651, 844, 217, 206, 725} \[ -\frac {d^2 \sqrt {a+c x^2}}{e (d+e x) \left (a e^2+c d^2\right )}+\frac {d \left (2 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 844
Rule 1651
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x)^2 \sqrt {a+c x^2}} \, dx &=-\frac {d^2 \sqrt {a+c x^2}}{e \left (c d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {a d-\frac {\left (c d^2+a e^2\right ) x}{e}}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2}\\ &=-\frac {d^2 \sqrt {a+c x^2}}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\int \frac {1}{\sqrt {a+c x^2}} \, dx}{e^2}-\frac {\left (d \left (2 a+\frac {c d^2}{e^2}\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2}\\ &=-\frac {d^2 \sqrt {a+c x^2}}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{e^2}+\frac {\left (d \left (2 a+\frac {c d^2}{e^2}\right )\right ) \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^2 \sqrt {a+c x^2}}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}+\frac {d \left (2 a+\frac {c d^2}{e^2}\right ) \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 = 0.32, size = 172, normalized size = 1.26 \[ \frac {d \left (\frac {\left (2 a e^2+c d^2\right ) \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {d e \sqrt {a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}\right )-\frac {\left (2 a d e^2+c d^3\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}+\frac {\log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{\sqrt {c}}}{e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 13.03, size = 1260, normalized size = 9.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 368, normalized size = 2.69 \[ -\frac {c \,d^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{3}}-\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{2}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) e^{2}}+\frac {2 d \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{3}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}\, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 171, normalized size = 1.25 \[ -\frac {\sqrt {c x^{2} + a} d^{2}}{c d^{2} e^{2} x + a e^{4} x + c d^{3} e + a d e^{3}} + \frac {\operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c} e^{2}} + \frac {c d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{5}} - \frac {2 \, d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{\sqrt {a + \frac {c d^{2}}{e^{2}}} e^{3}} \]
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 \[ \int \frac {x^2}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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