Optimal. Leaf size=94 \[ -\frac {a+b \sin ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^5 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2}}{12 d e^5 (c+d x)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4805, 12, 4627, 271, 264} \[ -\frac {a+b \sin ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^5 (c+d x)}-\frac {b \sqrt {1-(c+d x)^2}}{12 d e^5 (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^5} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{e^5 x^5} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^5} \, dx,x,c+d x\right )}{d e^5}\\ &=-\frac {a+b \sin ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d e^5}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{12 d e^5 (c+d x)^3}-\frac {a+b \sin ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{6 d e^5}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{12 d e^5 (c+d x)^3}-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^5 (c+d x)}-\frac {a+b \sin ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 63, normalized size = 0.67 \[ -\frac {3 \left (a+b \sin ^{-1}(c+d x)\right )+b (c+d x) \sqrt {1-(c+d x)^2} \left (2 (c+d x)^2+1\right )}{12 d e^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 193, normalized size = 2.05 \[ \frac {3 \, a d^{4} x^{4} + 12 \, a c d^{3} x^{3} + 18 \, a c^{2} d^{2} x^{2} + 12 \, a c^{3} d x - 3 \, b c^{4} \arcsin \left (d x + c\right ) - {\left (2 \, b c^{4} d^{3} x^{3} + 6 \, b c^{5} d^{2} x^{2} + 2 \, b c^{7} + b c^{5} + {\left (6 \, b c^{6} + b c^{4}\right )} d x\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{12 \, {\left (c^{4} d^{5} e^{5} x^{4} + 4 \, c^{5} d^{4} e^{5} x^{3} + 6 \, c^{6} d^{3} e^{5} x^{2} + 4 \, c^{7} d^{2} e^{5} x + c^{8} d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 432, normalized size = 4.60 \[ -\frac {1}{192} \, d {\left (\frac {18 \, b \arcsin \left (d x + c\right ) e^{\left (-7\right )}}{d^{2}} + \frac {3 \, {\left (d x + c\right )}^{4} b \arcsin \left (d x + c\right ) e^{\left (-7\right )}}{d^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4}} + \frac {12 \, {\left (d x + c\right )}^{2} b \arcsin \left (d x + c\right ) e^{\left (-7\right )}}{d^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {12 \, b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right ) e^{\left (-7\right )}}{{\left (d x + c\right )}^{2} d^{2}} + \frac {3 \, b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4} \arcsin \left (d x + c\right ) e^{\left (-7\right )}}{{\left (d x + c\right )}^{4} d^{2}} + \frac {18 \, a e^{\left (-7\right )}}{d^{2}} + \frac {3 \, {\left (d x + c\right )}^{4} a e^{\left (-7\right )}}{d^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4}} - \frac {2 \, {\left (d x + c\right )}^{3} b e^{\left (-7\right )}}{d^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} + \frac {12 \, {\left (d x + c\right )}^{2} a e^{\left (-7\right )}}{d^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {18 \, {\left (d x + c\right )} b e^{\left (-7\right )}}{d^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} + \frac {18 \, b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} e^{\left (-7\right )}}{{\left (d x + c\right )} d^{2}} + \frac {12 \, a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} e^{\left (-7\right )}}{{\left (d x + c\right )}^{2} d^{2}} + \frac {2 \, b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3} e^{\left (-7\right )}}{{\left (d x + c\right )}^{3} d^{2}} + \frac {3 \, a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4} e^{\left (-7\right )}}{{\left (d x + c\right )}^{4} d^{2}}\right )} e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.89 \[ \frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{12 \left (d x +c \right )^{3}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{6 \left (d x +c \right )}\right )}{e^{5}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 263, normalized size = 2.80 \[ \frac {1}{12} \, b {\left (\frac {{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 2 \, c^{4} + {\left (12 \, c^{2} d^{2} - d^{2}\right )} x^{2} - c^{2} + 2 \, {\left (4 \, c^{3} d - c d\right )} x - 1\right )} d}{{\left (d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}\right )} \sqrt {d x + c + 1} \sqrt {-d x - c + 1}} - \frac {3 \, \arcsin \left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} - \frac {a}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^5} \,d x \]
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 \[ \frac {\int \frac {a}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx + \int \frac {b \operatorname {asin}{\left (c + d x \right )}}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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