Optimal. Leaf size=135 \[ -\frac {a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^3 d \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 e \left (c^2 d^2-e^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, 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 = {4743, 731, 725, 204} \[ -\frac {a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^3 d \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 e \left (c^2 d^2-e^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 725
Rule 731
Rule 4743
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 ) \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 \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*}
________________________________________________________________________________________
Mathematica [C] time = 0.41, size = 207, normalized size = 1.53 \[ \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}+i c^2 d x+i e\right )}{b c^3 d (d+e x)}\right )\right )}{e (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}-\frac {b \sin ^{-1}(c x)}{e (d+e x)^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.80, size = 673, normalized size = 4.99 \[ \left [-\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{2} e^{2} + 2 \, a e^{4} - {\left (b c^{3} d e^{2} x^{2} + 2 \, b c^{3} d^{2} e x + b c^{3} d^{3}\right )} \sqrt {-c^{2} d^{2} + e^{2}} \log \left (\frac {2 \, c^{2} d e x - c^{2} d^{2} + {\left (2 \, c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} + 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1} + 2 \, e^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (b c^{4} d^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \arcsin \left (c x\right ) - 2 \, {\left (b c^{3} d^{3} e - b c d e^{3} + {\left (b c^{3} d^{2} e^{2} - b c e^{4}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{4} e^{3} + d^{2} e^{5} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{2} e^{5} + e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{3} e^{4} + d e^{6}\right )} x\right )}}, -\frac {a c^{4} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a e^{4} - {\left (b c^{3} d e^{2} x^{2} + 2 \, b c^{3} d^{2} e x + b c^{3} d^{3}\right )} \sqrt {c^{2} d^{2} - e^{2}} \arctan \left (\frac {\sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1}}{c^{2} d^{2} - {\left (c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} - e^{2}}\right ) + {\left (b c^{4} d^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \arcsin \left (c x\right ) - {\left (b c^{3} d^{3} e - b c d e^{3} + {\left (b c^{3} d^{2} e^{2} - b c e^{4}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{4} e^{3} + d^{2} e^{5} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{2} e^{5} + e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{3} e^{4} + d e^{6}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 301, normalized size = 2.23 \[ -\frac {c^{2} a}{2 \left (c e x +d c \right )^{2} e}-\frac {c^{2} b \arcsin \left (c x \right )}{2 \left (c e x +d c \right )^{2} e}+\frac {c^{2} b \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {c^{3} b d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________