Optimal. Leaf size=102 \[ -\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{15 d e^{7/2}}-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 102, 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, 4627, 325, 329, 221} \[ -\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}+\frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{15 d e^{7/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 325
Rule 329
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{(e x)^{5/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d e}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{15 d e^3}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{15 d e^4}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{15 d e^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 59, normalized size = 0.58 \[ \frac {-6 \left (a+b \sin ^{-1}(c+d x)\right )-4 b (c+d x) \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};(c+d x)^2\right )}{15 d e (e (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 169, normalized size = 1.66 \[ \frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{15 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{15 e^{2} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left ({\left (d x + c\right )}^{\frac {5}{2}} b \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + {\left (a d^{2} x^{2} + {\left (b d^{3} e^{4} x^{2} + 2 \, b c d^{2} e^{4} x + b c^{2} d e^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} \int \frac {\sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1}}{d^{5} e^{4} x^{5} + 5 \, c d^{4} e^{4} x^{4} + {\left (10 \, c^{2} - 1\right )} d^{3} e^{4} x^{3} + {\left (10 \, c^{3} - 3 \, c\right )} d^{2} e^{4} x^{2} + {\left (5 \, c^{4} - 3 \, c^{2}\right )} d e^{4} x + {\left (c^{5} - c^{3}\right )} e^{4}}\,{d x} + 2 \, a c d x + a c^{2}\right )} \sqrt {d x + c}\right )}}{5 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )} {\left (d x + c\right )}^{3} \sqrt {e}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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