Optimal. Leaf size=117 \[ \frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}+\frac {4 b \sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}}{25 d}+\frac {12 b e \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {c+d x}} \]
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Rubi [A] time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4805, 4627, 321, 320, 318, 424} \[ \frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}+\frac {4 b \sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}}{25 d}+\frac {12 b e \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 318
Rule 320
Rule 321
Rule 424
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^{3/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{3/2} \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d e}\\ &=\frac {4 b (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}-\frac {(6 b e) \operatorname {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{25 d}\\ &=\frac {4 b (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}-\frac {\left (6 b e \sqrt {e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{25 d \sqrt {c+d x}}\\ &=\frac {4 b (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}+\frac {\left (12 b e \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 )}{25 d \sqrt {c+d x}}\\ &=\frac {4 b (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right )}{5 d e}+\frac {12 b e \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 87, normalized size = 0.74 \[ \frac {2 (e (c+d x))^{3/2} \left (5 a c+5 a d x-2 b \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )+2 b \sqrt {1-(c+d x)^2}+5 b c \sin ^{-1}(c+d x)+5 b d x \sin ^{-1}(c+d x)\right )}{25 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a d e x + a c e + {\left (b d e x + b c e\right )} \arcsin \left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]
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 \[ \int {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 194, normalized size = 1.66 \[ \frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{5}-\frac {3 e^{3} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{5 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{5 e}\right )}{d e} \]
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 \[ \frac {2 \, {\left (b d^{2} e x^{2} + 2 \, b c d e x + b c^{2} e\right )} \sqrt {d x + c} \sqrt {e} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {5}{2}} a e + d \int \frac {{\left (b d^{2} e x^{2} + 2 \, b c d e x + b c^{2} e\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x}\right )} \sqrt {e}}{5 \, d} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.22, size = 156, normalized size = 1.33 \[ a c e \left (\begin {cases} x \sqrt {c e} & \text {for}\: d = 0 \\0 & \text {for}\: e = 0 \\\frac {2 \left (c e + d e x\right )^{\frac {3}{2}}}{3 d e} & \text {otherwise} \end {cases}\right ) - \frac {2 a c \left (c e + d e x\right )^{\frac {3}{2}}}{3 d} + \frac {2 a \left (c e + d e x\right )^{\frac {5}{2}}}{5 d e} + \frac {2 b \left (c e + d e x\right )^{\frac {5}{2}} \operatorname {asin}{\left (c + d x \right )}}{5 d e} - \frac {b \left (c e + d e x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {\left (c e + d e x\right )^{2} e^{2 i \pi }}{e^{2}}} \right )}}{5 d e^{2} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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