Optimal. Leaf size=135 \[ \frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {ArcSin}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e \left (c^2 d^2-e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4827, 745, 739,
210} \begin {gather*} -\frac {a+b \text {ArcSin}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 739
Rule 745
Rule 4827
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}+\frac {(b c) \int \frac {1}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (b c^3 d\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e \left (c^2 d^2-e^2\right )}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}-\frac {\left (b c^3 d\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 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \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 \left (c^2 d^2-e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.25, size = 207, normalized size = 1.53 \begin {gather*} \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b \text {ArcSin}(c x)}{e (d+e x)^2}-\frac {i b c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{b c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \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. \(304\) vs.
\(2(124)=248\).
time = 0.32, size = 305, normalized size = 2.26
method | result | size |
derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +d c \right )^{2} e}-\frac {b \,c^{3} \arcsin \left (c x \right )}{2 \left (c e x +d c \right )^{2} e}+\frac {b \,c^{3} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{4} d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\) | \(305\) |
default | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +d c \right )^{2} e}-\frac {b \,c^{3} \arcsin \left (c x \right )}{2 \left (c e x +d c \right )^{2} e}+\frac {b \,c^{3} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{4} d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\) | \(305\) |
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. 314 vs.
\(2 (119) = 238\).
time = 2.45, size = 657, normalized size = 4.87 \begin {gather*} \left [-\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{2} e^{2} - {\left (b c^{3} d x^{2} e^{2} + 2 \, b c^{3} d^{2} x e + b c^{3} d^{3}\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^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \arcsin \left (c x\right ) + 2 \, a e^{4} - 2 \, {\left (b c^{3} d^{2} x e^{2} + b c^{3} d^{3} e - b c x e^{4} - b c d e^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, {\left (2 \, c^{4} d^{5} x e^{2} + c^{4} d^{6} e - 4 \, c^{2} d^{3} x e^{4} + x^{2} e^{7} + 2 \, d x e^{6} - {\left (2 \, c^{2} d^{2} x^{2} - d^{2}\right )} e^{5} + {\left (c^{4} d^{4} x^{2} - 2 \, c^{2} d^{4}\right )} e^{3}\right )}}, -\frac {a c^{4} d^{4} - 2 \, a c^{2} d^{2} e^{2} - {\left (b c^{3} d x^{2} e^{2} + 2 \, b c^{3} d^{2} x e + b c^{3} d^{3}\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^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \arcsin \left (c x\right ) + a e^{4} - {\left (b c^{3} d^{2} x e^{2} + b c^{3} d^{3} e - b c x e^{4} - b c d e^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (2 \, c^{4} d^{5} x e^{2} + c^{4} d^{6} e - 4 \, c^{2} d^{3} x e^{4} + x^{2} e^{7} + 2 \, d x e^{6} - {\left (2 \, c^{2} d^{2} x^{2} - d^{2}\right )} e^{5} + {\left (c^{4} d^{4} x^{2} - 2 \, c^{2} d^{4}\right )} e^{3}\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 {a + b \operatorname {asin}{\left (c 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 = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x\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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