Optimal. Leaf size=136 \[ \frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d}+\frac {20 b e^2 \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}}{147 d}+\frac {4 b \sqrt {1-(c+d x)^2} (e (c+d x))^{5/2}}{49 d} \]
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Rubi [A] time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}+\frac {20 b e^2 \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}}{147 d}-\frac {20 b e^{5/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d}+\frac {4 b \sqrt {1-(c+d x)^2} (e (c+d x))^{5/2}}{49 d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 321
Rule 329
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^{5/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{5/2} \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{7/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{7 d e}\\ &=\frac {4 b (e (c+d x))^{5/2} \sqrt {1-(c+d x)^2}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}-\frac {(10 b e) \operatorname {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{49 d}\\ &=\frac {20 b e^2 \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{147 d}+\frac {4 b (e (c+d x))^{5/2} \sqrt {1-(c+d x)^2}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}-\frac {\left (10 b e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{147 d}\\ &=\frac {20 b e^2 \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{147 d}+\frac {4 b (e (c+d x))^{5/2} \sqrt {1-(c+d x)^2}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}-\frac {\left (20 b e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{147 d}\\ &=\frac {20 b e^2 \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{147 d}+\frac {4 b (e (c+d x))^{5/2} \sqrt {1-(c+d x)^2}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 115, normalized size = 0.85 \[ \frac {2 (e (c+d x))^{5/2} \left (21 a (c+d x)^3-10 b \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )+6 b \sqrt {1-(c+d x)^2} (c+d x)^2+10 b \sqrt {1-(c+d x)^2}+21 b (c+d x)^3 \sin ^{-1}(c+d x)\right )}{147 d (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a d^{2} e^{2} x^{2} + 2 \, a c d e^{2} x + a c^{2} e^{2} + {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + b c^{2} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]
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 \[ \int {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 206, normalized size = 1.51 \[ \frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {\frac {d e x +c e}{e}+1}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{21 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{d e} \]
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 \[ \frac {2 \, {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \sqrt {d x + c} \sqrt {e} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {7}{2}} a e^{2} + d \int \frac {{\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x}\right )} \sqrt {e}}{7 \, d} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^{5/2}\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 133.13, size = 163, normalized size = 1.20 \[ a c^{2} e^{2} \left (\begin {cases} x \sqrt {c e} & \text {for}\: d = 0 \\0 & \text {for}\: e = 0 \\\frac {2 \left (c e + d e x\right )^{\frac {3}{2}}}{3 d e} & \text {otherwise} \end {cases}\right ) - \frac {2 a c^{2} e \left (c e + d e x\right )^{\frac {3}{2}}}{3 d} + \frac {2 a \left (c e + d e x\right )^{\frac {7}{2}}}{7 d e} + \frac {2 b \left (c e + d e x\right )^{\frac {7}{2}} \operatorname {asin}{\left (c + d x \right )}}{7 d e} - \frac {b \left (c e + d e x\right )^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {\left (c e + d e x\right )^{2} e^{2 i \pi }}{e^{2}}} \right )}}{7 d e^{2} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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