Optimal. Leaf size=159 \[ -\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac {12 b \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{35 d e^5 \sqrt {c+d x}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4805, 4627, 325, 320, 318, 424} \[ -\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}+\frac {12 b \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{35 d e^5 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 318
Rule 320
Rule 325
Rule 424
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{9/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{(e x)^{9/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{(e x)^{7/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {1}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{35 d e^3}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac {(6 b) \operatorname {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{35 d e^5}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac {\left (6 b \sqrt {e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{35 d e^5 \sqrt {c+d x}}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac {\left (12 b \sqrt {e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )}{35 d e^5 \sqrt {c+d x}}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac {12 b \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{35 d e^5 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 66, normalized size = 0.42 \[ -\frac {2 \sqrt {e (c+d x)} \left (5 \left (a+b \sin ^{-1}(c+d x)\right )+2 b (c+d x) \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right )\right )}{35 d e^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}}{d^{5} e^{5} x^{5} + 5 \, c d^{4} e^{5} x^{4} + 10 \, c^{2} d^{3} e^{5} x^{3} + 10 \, c^{3} d^{2} e^{5} x^{2} + 5 \, c^{4} d e^{5} x + c^{5} e^{5}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 225, normalized size = 1.42 \[ \frac {-\frac {2 a}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 \left (d e x +c e \right )^{\frac {5}{2}}}-\frac {6 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 e^{2} \sqrt {d e x +c e}}+\frac {6 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{35 e^{3} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e} \]
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 \[ -\frac {2 \, {\left ({\left (d x + c\right )}^{\frac {7}{2}} b \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + {\left (a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + {\left (b d^{4} e^{5} x^{3} + 3 \, b c d^{3} e^{5} x^{2} + 3 \, b c^{2} d^{2} e^{5} x + b c^{3} d e^{5}\right )} {\left (d x + c\right )}^{\frac {7}{2}} \int \frac {\sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1}}{d^{6} e^{5} x^{6} + 6 \, c d^{5} e^{5} x^{5} + {\left (15 \, c^{2} - 1\right )} d^{4} e^{5} x^{4} + 4 \, {\left (5 \, c^{3} - c\right )} d^{3} e^{5} x^{3} + 3 \, {\left (5 \, c^{4} - 2 \, c^{2}\right )} d^{2} e^{5} x^{2} + 2 \, {\left (3 \, c^{5} - 2 \, c^{3}\right )} d e^{5} x + {\left (c^{6} - c^{4}\right )} e^{5}}\,{d x} + 3 \, a c^{2} d x + a c^{3}\right )} \sqrt {d x + c}\right )}}{7 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )} {\left (d x + c\right )}^{4} \sqrt {e}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{9/2}} \,d x \]
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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