Optimal. Leaf size=130 \[ \frac {16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};(c+d x)^2\right )}{105 d e^3}-\frac {8 b (e (c+d x))^{5/2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{15 d e^2}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d e} \]
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Rubi [A] time = 0.20, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4805, 4627, 4711} \[ \frac {16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};(c+d x)^2\right )}{105 d e^3}-\frac {8 b (e (c+d x))^{5/2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{15 d e^2}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d e} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4711
Rule 4805
Rubi steps
\begin {align*} \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {e x} \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {(e x)^{3/2} \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d e}-\frac {8 b (e (c+d x))^{5/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};(c+d x)^2\right )}{15 d e^2}+\frac {16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};(c+d x)^2\right )}{105 d e^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 107, normalized size = 0.82 \[ \frac {2 (e (c+d x))^{3/2} \left (8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};(c+d x)^2\right )+7 \left (a+b \sin ^{-1}(c+d x)\right ) \left (5 \left (a+b \sin ^{-1}(c+d x)\right )-4 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};(c+d x)^2\right )\right )\right )}{105 d e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2}\right )} \sqrt {d e x + c e}, x\right ) \]
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 \[ \int \sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x +c e}\, \left (a +b \arcsin \left (d x +c \right )\right )^{2}\, dx \]
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 {4 \, {\left (b^{2} d x + b^{2} 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 (12 \, a b d^{2} \sqrt {e} \int \frac {\sqrt {d x + c} x^{2} \arctan \left (\frac {d x}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}} + \frac {c}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} + 24 \, a b c d \sqrt {e} \int \frac {\sqrt {d x + c} x \arctan \left (\frac {d x}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}} + \frac {c}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} + 12 \, a b c^{2} \sqrt {e} \int \frac {\sqrt {d x + c} \arctan \left (\frac {d x}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}} + \frac {c}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} + \frac {3 \, a^{2} c^{2} \sqrt {e} {\left (2 \, \arctan \left (\sqrt {d x + c}\right ) - \log \left (\sqrt {d x + c} + 1\right ) + \log \left (\sqrt {d x + c} - 1\right )\right )}}{d} + 8 \, b^{2} d \sqrt {e} \int \frac {\sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1} x \arctan \left (\frac {d x}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}} + \frac {c}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} + 8 \, b^{2} c \sqrt {e} \int \frac {\sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {-d x - c + 1} \arctan \left (\frac {d x}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}} + \frac {c}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} - \frac {6 \, {\left (2 \, {\left (c + 1\right )} \arctan \left (\sqrt {d x + c}\right ) - {\left (c - 1\right )} \log \left (\sqrt {d x + c} + 1\right ) + {\left (c - 1\right )} \log \left (\sqrt {d x + c} - 1\right ) - 4 \, \sqrt {d x + c}\right )} a^{2} c \sqrt {e}}{d} - 12 \, a b \sqrt {e} \int \frac {\sqrt {d x + c} \arctan \left (\frac {d x}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}} + \frac {c}{\sqrt {d x + c + 1} \sqrt {-d x - c + 1}}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} + \frac {{\left (6 \, {\left (c^{2} + 2 \, c + 1\right )} \arctan \left (\sqrt {d x + c}\right ) - 3 \, {\left (c^{2} - 2 \, c + 1\right )} \log \left (\sqrt {d x + c} + 1\right ) + 3 \, {\left (c^{2} - 2 \, c + 1\right )} \log \left (\sqrt {d x + c} - 1\right ) + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} - 24 \, \sqrt {d x + c} c\right )} a^{2} \sqrt {e}}{d} - \frac {3 \, a^{2} \sqrt {e} {\left (2 \, \arctan \left (\sqrt {d x + c}\right ) - \log \left (\sqrt {d x + c} + 1\right ) + \log \left (\sqrt {d x + c} - 1\right )\right )}}{d}\right )} d}{6 \, 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 \[ \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
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 \[ \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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