Optimal. Leaf size=99 \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}+\frac {4 b \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}}{9 d}-\frac {4 b \sqrt {e} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d} \]
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Rubi [A] time = 0.08, antiderivative size = 99, 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, 321, 329, 221} \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}+\frac {4 b \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}}{9 d}-\frac {4 b \sqrt {e} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 321
Rule 329
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {e x} \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e}\\ &=\frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=\frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{9 d}\\ &=\frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {e} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 87, normalized size = 0.88 \[ \frac {2 \sqrt {e (c+d x)} \left (3 a c+3 a d x-2 b \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )+2 b \sqrt {1-(c+d x)^2}+3 b c \sin ^{-1}(c+d x)+3 b d x \sin ^{-1}(c+d x)\right )}{9 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}, x\right ) \]
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 \[ \int \sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 172, normalized size = 1.74 \[ \frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (-\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}+\frac {e^{2} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{3 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e} \]
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 \[ \frac {2 \, {\left (b d x + b c\right )} \sqrt {d x + c} \sqrt {e} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} a + d \int \frac {{\left (b d x + b c\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x}\right )} \sqrt {e}}{3 \, d} \]
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 \[ \int \sqrt {c\,e+d\,e\,x}\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.40, size = 104, normalized size = 1.05 \[ \frac {2 a \left (c e + d e x\right )^{\frac {3}{2}}}{3 d e} + \frac {2 b \left (c e + d e x\right )^{\frac {3}{2}} \operatorname {asin}{\left (c + d x \right )}}{3 d e} - \frac {b \left (c e + d e x\right )^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {\left (c e + d e x\right )^{2} e^{2 i \pi }}{e^{2}}} \right )}}{3 d e^{2} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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