Optimal. Leaf size=268 \[ \frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {c e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^4 \sqrt {a e^2+c d^2}}+\frac {e^4 \sqrt {a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {961, 266, 51, 63, 208, 264, 731, 725, 206} \begin {gather*} \frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^2}+\frac {e^4 \sqrt {a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )}+\frac {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^4 \sqrt {a e^2+c d^2}}+\frac {c e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 208
Rule 264
Rule 266
Rule 725
Rule 731
Rule 961
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx &=\int \left (\frac {1}{d^2 x^3 \sqrt {a+c x^2}}-\frac {2 e}{d^3 x^2 \sqrt {a+c x^2}}+\frac {3 e^2}{d^4 x \sqrt {a+c x^2}}-\frac {e^3}{d^3 (d+e x)^2 \sqrt {a+c x^2}}-\frac {3 e^3}{d^4 (d+e x) \sqrt {a+c x^2}}\right ) \, dx\\ &=\frac {\int \frac {1}{x^3 \sqrt {a+c x^2}} \, dx}{d^2}-\frac {(2 e) \int \frac {1}{x^2 \sqrt {a+c x^2}} \, dx}{d^3}+\frac {\left (3 e^2\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{d^4}-\frac {\left (3 e^3\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^4}-\frac {e^3 \int \frac {1}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{d^3}\\ &=\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^4}+\frac {\left (3 e^3\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 )}{d^4}-\frac {\left (c e^3\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^2 \left (c d^2+a e^2\right )}\\ &=-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}-\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a d^2}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^4}+\frac {\left (c e^3\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 )}{d^2 \left (c d^2+a e^2\right )}\\ &=-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {c e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a d^2}\\ &=-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {c e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 229, normalized size = 0.85 \begin {gather*} \frac {\frac {\left (c d^2-6 a e^2\right ) \log \left (\sqrt {a} \sqrt {a+c x^2}+a\right )}{a^{3/2}}+\frac {\log (x) \left (6 a e^2-c d^2\right )}{a^{3/2}}+d \sqrt {a+c x^2} \left (\frac {2 e^4}{(d+e x) \left (a e^2+c d^2\right )}-\frac {d-4 e x}{a x^2}\right )+\frac {2 e^3 \left (3 a e^2+4 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 {2 e^3 \left (3 a e^2+4 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{2 d^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.78, size = 248, normalized size = 0.93 \begin {gather*} \frac {\left (6 a e^2-c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^4}-\frac {2 \sqrt {-a e^2-c d^2} \left (3 a e^5+4 c d^2 e^3\right ) \tan ^{-1}\left (\frac {-e \sqrt {a+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {-a e^2-c d^2}}\right )}{d^4 \left (a e^2+c d^2\right )^2}+\frac {\sqrt {a+c x^2} \left (-a d^2 e^2+3 a d e^3 x+6 a e^4 x^2-c d^4+3 c d^3 e x+4 c d^2 e^2 x^2\right )}{2 a d^3 x^2 (d+e x) \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 1867, normalized size = 6.97
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 452, normalized size = 1.69 \begin {gather*} \frac {c \,e^{2} \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}}}\, d^{2}}+\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}}}\, e^{3}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) d^{3}}+\frac {3 e^{2} \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}}}\, d^{4}}-\frac {3 e^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{\sqrt {a}\, d^{4}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}} d^{2}}+\frac {2 \sqrt {c \,x^{2}+a}\, e}{a \,d^{3} x}-\frac {\sqrt {c \,x^{2}+a}}{2 a \,d^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{3} \sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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