Optimal. Leaf size=61 \[ \frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4805, 4627, 329, 221} \[ \frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 329
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{d e^2}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {4 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 54, normalized size = 0.89 \[ -\frac {2 \left (a-2 b (c+d x) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )+b \sin ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 132, normalized size = 2.16 \[ \frac {-\frac {2 a}{\sqrt {d e x +c e}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{d e} \]
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 \[ -\frac {2 \, {\left (\sqrt {d x + c} b \sqrt {e} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + {\left (\sqrt {d x + c} b d \int \frac {\sqrt {-d x - c + 1}}{\sqrt {d x + c + 1} \sqrt {d x + c} d x + \sqrt {d x + c + 1} \sqrt {d x + c} c - \sqrt {d x + c + 1} \sqrt {d x + c}}\,{d x} + a\right )} \sqrt {d x + c} \sqrt {e}\right )}}{{\left (d x + c\right )} d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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