Optimal. Leaf size=147 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {e (a e+c d x)}{a d \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}+\frac {1}{a d \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 10, 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, 741, 12, 725, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {e (a e+c d x)}{a d \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}+\frac {1}{a d \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 206
Rule 208
Rule 266
Rule 725
Rule 741
Rule 961
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\int \left (\frac {1}{d x \left (a+c x^2\right )^{3/2}}-\frac {e}{d (d+e x) \left (a+c x^2\right )^{3/2}}\right ) \, dx\\ &=\frac {\int \frac {1}{x \left (a+c x^2\right )^{3/2}} \, dx}{d}-\frac {e \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{d}\\ &=-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d}-\frac {e \int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a d \left (c d^2+a e^2\right )}\\ &=\frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d}-\frac {e^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d \left (c d^2+a e^2\right )}\\ &=\frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c d}+\frac {e^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 )}{d \left (c d^2+a e^2\right )}\\ &=\frac {1}{a d \sqrt {a+c x^2}}-\frac {e (a e+c d x)}{a d \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {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 \left (c d^2+a e^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 132, normalized size = 0.90 \[ \frac {-\frac {e (a e+c d x)}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c x^2}{a}+1\right )}{a \sqrt {a+c x^2}}}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.55, size = 1325, normalized size = 9.01 \[ \text {result too large to display} \]
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 = 318, normalized size = 2.16 \[ -\frac {c e x}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a}+\frac {e^{2} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d}-\frac {e^{2}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d}-\frac {\ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {3}{2}} d}+\frac {1}{\sqrt {c \,x^{2}+a}\, a d} \]
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 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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