Optimal. Leaf size=202 \[ \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \text {ArcSin}(c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 (a+b \text {ArcSin}(c x))}{2 (e f-d g) (d+e x)^2}-\frac {b c \left (2 e^2 g-c^2 d (e f+d g)\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.26, 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, 4837, 12,
1665, 858, 222, 739, 210} \begin {gather*} -\frac {(f+g x)^2 (a+b \text {ArcSin}(c x))}{2 (d+e x)^2 (e f-d g)}+\frac {b g^2 \text {ArcSin}(c x)}{2 e^2 (e f-d g)}-\frac {b c \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 e^2 g-c^2 d (d g+e f)\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 210
Rule 222
Rule 739
Rule 858
Rule 1665
Rule 4837
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 ) \text {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 = 0.35, size = 263, normalized size = 1.30 \begin {gather*} \frac {\frac {a (-e f+d g)}{(d+e x)^2}-\frac {2 a g}{d+e x}-\frac {b c e (e f-d g) \sqrt {1-c^2 x^2}}{\left (-c^2 d^2+e^2\right ) (d+e x)}-\frac {b (d g+e (f+2 g x)) \text {ArcSin}(c x)}{(d+e x)^2}+\frac {b c \left (-2 e^2 g+c^2 d (e f+d g)\right ) \log (d+e x)}{(c d-e) (c d+e) \sqrt {-c^2 d^2+e^2}}+\frac {b c \left (-2 e^2 g+c^2 d (e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e) (c d+e) \sqrt {-c^2 d^2+e^2}}}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(799\) vs.
\(2(186)=372\).
time = 0.13, size = 800, normalized size = 3.96
method | result | size |
derivativedivides | \(\frac {a \,c^{2} \left (\frac {c \left (d g -e f \right )}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {g}{e^{2} \left (c e x +d c \right )}\right )+\frac {b \,c^{3} \arcsin \left (c x \right ) d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {b \,c^{3} \arcsin \left (c x \right ) f}{2 e \left (c e x +d c \right )^{2}}-\frac {b \,c^{2} \arcsin \left (c x \right ) g}{e^{2} \left (c e x +d c \right )}-\frac {b \,c^{3} \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 {b \,c^{3} \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 {b \,c^{4} 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 {b \,c^{4} 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 {b \,c^{2} 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}}}}}{c}\) | \(800\) |
default | \(\frac {a \,c^{2} \left (\frac {c \left (d g -e f \right )}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {g}{e^{2} \left (c e x +d c \right )}\right )+\frac {b \,c^{3} \arcsin \left (c x \right ) d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {b \,c^{3} \arcsin \left (c x \right ) f}{2 e \left (c e x +d c \right )^{2}}-\frac {b \,c^{2} \arcsin \left (c x \right ) g}{e^{2} \left (c e x +d c \right )}-\frac {b \,c^{3} \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 {b \,c^{3} \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 {b \,c^{4} 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 {b \,c^{4} 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 {b \,c^{2} 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}}}}}{c}\) | \(800\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 557 vs.
\(2 (185) = 370\).
time = 8.86, size = 1145, normalized size = 5.67 \begin {gather*} \left [-\frac {2 \, a c^{4} d^{5} g - 4 \, a c^{2} d^{3} g e^{2} + 2 \, a d g e^{4} + {\left (b c^{3} d^{4} g - 2 \, b c g x^{2} e^{4} + {\left (b c^{3} d f x^{2} - 4 \, b c d g x\right )} e^{3} + {\left (b c^{3} d^{2} g x^{2} + 2 \, b c^{3} d^{2} f x - 2 \, b c d^{2} g\right )} e^{2} + {\left (2 \, b c^{3} d^{3} g x + b c^{3} d^{3} f\right )} e\right )} \sqrt {-c^{2} d^{2} + e^{2}} \log \left (\frac {2 \, c^{4} d^{2} x^{2} + 2 \, c^{2} d x e - c^{2} d^{2} - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1} - {\left (c^{2} x^{2} - 2\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (b c^{4} d^{5} g - 2 \, b c^{2} d^{3} g e^{2} + b d g e^{4} + {\left (2 \, b g x + b f\right )} e^{5} - 2 \, {\left (2 \, b c^{2} d^{2} g x + b c^{2} d^{2} f\right )} e^{3} + {\left (2 \, b c^{4} d^{4} g x + b c^{4} d^{4} f\right )} e\right )} \arcsin \left (c x\right ) + 2 \, {\left (2 \, a g x + a f\right )} e^{5} - 4 \, {\left (2 \, a c^{2} d^{2} g x + a c^{2} d^{2} f\right )} e^{3} + 2 \, {\left (2 \, a c^{4} d^{4} g x + a c^{4} d^{4} f\right )} e + 2 \, {\left (b c^{3} d^{4} g e + b c f x e^{5} - {\left (b c d g x - b c d f\right )} e^{4} - {\left (b c^{3} d^{2} f x + b c d^{2} g\right )} e^{3} + {\left (b c^{3} d^{3} g x - b c^{3} d^{3} f\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, {\left (2 \, c^{4} d^{5} x e^{3} + c^{4} d^{6} e^{2} - 4 \, c^{2} d^{3} x e^{5} + x^{2} e^{8} + 2 \, d x e^{7} - {\left (2 \, c^{2} d^{2} x^{2} - d^{2}\right )} e^{6} + {\left (c^{4} d^{4} x^{2} - 2 \, c^{2} d^{4}\right )} e^{4}\right )}}, -\frac {a c^{4} d^{5} g - 2 \, a c^{2} d^{3} g e^{2} + a d g e^{4} - {\left (b c^{3} d^{4} g - 2 \, b c g x^{2} e^{4} + {\left (b c^{3} d f x^{2} - 4 \, b c d g x\right )} e^{3} + {\left (b c^{3} d^{2} g x^{2} + 2 \, b c^{3} d^{2} f x - 2 \, b c d^{2} g\right )} e^{2} + {\left (2 \, b c^{3} d^{3} g x + b c^{3} d^{3} f\right )} e\right )} \sqrt {c^{2} d^{2} - e^{2}} \arctan \left (-\frac {\sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1}}{c^{4} d^{2} x^{2} - c^{2} d^{2} - {\left (c^{2} x^{2} - 1\right )} e^{2}}\right ) + {\left (b c^{4} d^{5} g - 2 \, b c^{2} d^{3} g e^{2} + b d g e^{4} + {\left (2 \, b g x + b f\right )} e^{5} - 2 \, {\left (2 \, b c^{2} d^{2} g x + b c^{2} d^{2} f\right )} e^{3} + {\left (2 \, b c^{4} d^{4} g x + b c^{4} d^{4} f\right )} e\right )} \arcsin \left (c x\right ) + {\left (2 \, a g x + a f\right )} e^{5} - 2 \, {\left (2 \, a c^{2} d^{2} g x + a c^{2} d^{2} f\right )} e^{3} + {\left (2 \, a c^{4} d^{4} g x + a c^{4} d^{4} f\right )} e + {\left (b c^{3} d^{4} g e + b c f x e^{5} - {\left (b c d g x - b c d f\right )} e^{4} - {\left (b c^{3} d^{2} f x + b c d^{2} g\right )} e^{3} + {\left (b c^{3} d^{3} g x - b c^{3} d^{3} f\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (2 \, c^{4} d^{5} x e^{3} + c^{4} d^{6} e^{2} - 4 \, c^{2} d^{3} x e^{5} + x^{2} e^{8} + 2 \, d x e^{7} - {\left (2 \, c^{2} d^{2} x^{2} - d^{2}\right )} e^{6} + {\left (c^{4} d^{4} x^{2} - 2 \, c^{2} d^{4}\right )} e^{4}\right )}}\right ] \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 {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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