Optimal. Leaf size=202 \[ -\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (d+e x)^2 (e f-d g)}+\frac {b c \sqrt {1-c^2 x^2} (e f-d g)}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c \left (2 e^2 g-c^2 d (d g+e f)\right ) \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b g^2 \sin ^{-1}(c x)}{2 e^2 (e f-d g)} \]
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Rubi [A] time = 0.36, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {37, 4753, 12, 1651, 844, 216, 725, 204} \[ -\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (d+e x)^2 (e f-d g)}+\frac {b c \sqrt {1-c^2 x^2} (e f-d g)}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c \left (2 e^2 g-c^2 d (d g+e f)\right ) \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b g^2 \sin ^{-1}(c x)}{2 e^2 (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 204
Rule 216
Rule 725
Rule 844
Rule 1651
Rule 4753
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^3} \, dx &=-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2}-(b c) \int -\frac {(f+g x)^2}{2 (e f-d g) (d+e x)^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2}+\frac {(b c) \int \frac {(f+g x)^2}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 (e f-d g)}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2}+\frac {(b c) \int \frac {c^2 d f^2-g (2 e f-d g)+\left (\frac {c^2 d^2}{e}-e\right ) g^2 x}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 \left (c^2 d^2-e^2\right ) (e f-d g)}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2}+\frac {\left (b c g^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 e^2 (e f-d g)}-\frac {\left (b c \left (2 e^2 g-c^2 d (e f+d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \sin ^{-1}(c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2}+\frac {\left (b c \left (2 e^2 g-c^2 d (e f+d g)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \sin ^{-1}(c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2}-\frac {b c \left (2 e^2 g-c^2 d (e f+d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 263, normalized size = 1.30 \[ \frac {\frac {a (d g-e f)}{(d+e x)^2}-\frac {2 a g}{d+e x}-\frac {b c e \sqrt {1-c^2 x^2} (e f-d g)}{\left (e^2-c^2 d^2\right ) (d+e x)}+\frac {b c \left (c^2 d (d g+e f)-2 e^2 g\right ) \log \left (\sqrt {1-c^2 x^2} \sqrt {e^2-c^2 d^2}+c^2 d x+e\right )}{(e-c d) (c d+e) \sqrt {e^2-c^2 d^2}}+\frac {b c \log (d+e x) \left (c^2 d (d g+e f)-2 e^2 g\right )}{(c d-e) (c d+e) \sqrt {e^2-c^2 d^2}}-\frac {b \sin ^{-1}(c x) (d g+e (f+2 g x))}{(d+e x)^2}}{2 e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 13.68, size = 1184, normalized size = 5.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 805, normalized size = 3.99 \[ -\frac {c a g}{e^{2} \left (c e x +d c \right )}+\frac {c^{2} a d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {c^{2} a f}{2 e \left (c e x +d c \right )^{2}}-\frac {c b \arcsin \left (c x \right ) g}{e^{2} \left (c e x +d c \right )}+\frac {c^{2} b \arcsin \left (c x \right ) d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {c^{2} b \arcsin \left (c x \right ) f}{2 e \left (c e x +d c \right )^{2}}-\frac {c^{2} b \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, d g}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}+\frac {c^{2} b \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, f}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}+\frac {c^{3} b \,d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right ) g}{2 e^{3} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{3} b d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right ) f}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c b g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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