Optimal. Leaf size=87 \[ -\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \]
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Rubi [A] time = 0.14, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4805, 12, 4627, 4681, 29} \[ -\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 4627
Rule 4681
Rule 4805
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 126, normalized size = 1.45 \[ -\frac {a \left (a+2 b (c+d x) \sqrt {-c^2-2 c d x-d^2 x^2+1}\right )+2 b \sin ^{-1}(c+d x) \left (a+b (c+d x) \sqrt {-c^2-2 c d x-d^2 x^2+1}\right )-2 b^2 (c+d x)^2 \log (c+d x)+b^2 \sin ^{-1}(c+d x)^2}{2 d e^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 146, normalized size = 1.68 \[ -\frac {b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right ) + 2 \, {\left (a b d x + a b c + {\left (b^{2} d x + b^{2} c\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 493, normalized size = 5.67 \[ -\frac {b^{2} \arcsin \left (d x + c\right )^{2} e^{\left (-3\right )}}{4 \, d} - \frac {{\left (d x + c\right )}^{2} b^{2} \arcsin \left (d x + c\right )^{2} e^{\left (-3\right )}}{8 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {b^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right )^{2} e^{\left (-3\right )}}{8 \, {\left (d x + c\right )}^{2} d} - \frac {a b \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{2 \, d} - \frac {{\left (d x + c\right )}^{2} a b \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{4 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{2 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{2 \, {\left (d x + c\right )} d} - \frac {a b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right ) e^{\left (-3\right )}}{4 \, {\left (d x + c\right )}^{2} d} + \frac {2 \, b^{2} e^{\left (-3\right )} \log \relax (2)}{d} - \frac {b^{2} e^{\left (-3\right )} \log \left (2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} + 2\right )}{d} + \frac {b^{2} e^{\left (-3\right )} \log \left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}{d} + \frac {b^{2} e^{\left (-3\right )} \log \left ({\left | d x + c \right |}\right )}{d} - \frac {a^{2} e^{\left (-3\right )}}{4 \, d} - \frac {{\left (d x + c\right )}^{2} a^{2} e^{\left (-3\right )}}{8 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} a b e^{\left (-3\right )}}{2 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {a b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} e^{\left (-3\right )}}{2 \, {\left (d x + c\right )} d} - \frac {a^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} e^{\left (-3\right )}}{8 \, {\left (d x + c\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 152, normalized size = 1.75 \[ -\frac {a^{2}}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsin \left (d x +c \right )^{2}}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{d \,e^{3} \left (d x +c \right )}+\frac {b^{2} \ln \left (d x +c \right )}{d \,e^{3}}-\frac {a b \arcsin \left (d x +c \right )}{d \,e^{3} \left (d x +c \right )^{2}}-\frac {a b \sqrt {1-\left (d x +c \right )^{2}}}{d \,e^{3} \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 236, normalized size = 2.71 \[ -{\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d \arcsin \left (d x + c\right )}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c\right )}{d e^{3}}\right )} b^{2} - a b {\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\arcsin \left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {b^{2} \arcsin \left (d x + c\right )^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {a^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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