Optimal. Leaf size=156 \[ \frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{405 d}+\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} (a+b \text {ArcSin}(c+d x))}{9 d e}+\frac {28 b e^3 \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{135 d \sqrt {c+d x}} \]
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Rubi [A]
time = 0.09, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 4723,
327, 326, 324, 435} \begin {gather*} \frac {2 (e (c+d x))^{9/2} (a+b \text {ArcSin}(c+d x))}{9 d e}+\frac {28 b e^3 \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{135 d \sqrt {c+d x}}+\frac {28 b e^2 \sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}}{405 d}+\frac {4 b \sqrt {1-(c+d x)^2} (e (c+d x))^{7/2}}{81 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 324
Rule 326
Rule 327
Rule 435
Rule 4723
Rule 4889
Rubi steps
\begin {align*} \int (c e+d e x)^{7/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int (e x)^{7/2} \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{9/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d e}\\ &=\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac {(14 b e) \text {Subst}\left (\int \frac {(e x)^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{81 d}\\ &=\frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{405 d}+\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac {\left (14 b e^3\right ) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{135 d}\\ &=\frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{405 d}+\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac {\left (14 b e^3 \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{135 d \sqrt {c+d x}}\\ &=\frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{405 d}+\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}+\frac {\left (28 b e^3 \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )}{135 d \sqrt {c+d x}}\\ &=\frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{405 d}+\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}+\frac {28 b e^3 \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{135 d \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.16, size = 115, normalized size = 0.74 \begin {gather*} \frac {2 (e (c+d x))^{7/2} \left (45 a (c+d x)^3+14 b \sqrt {1-(c+d x)^2}+10 b (c+d x)^2 \sqrt {1-(c+d x)^2}+45 b (c+d x)^3 \text {ArcSin}(c+d x)-14 b \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )\right )}{405 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.38, size = 228, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {9}{2}} a}{9}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}-\frac {7 e^{5} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{15 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\) | \(228\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {9}{2}} a}{9}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}-\frac {7 e^{5} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{15 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\) | \(228\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.68, size = 237, normalized size = 1.52 \begin {gather*} -\frac {2 \, {\left (42 \, \sqrt {-d^{3} e} b e^{3} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - {\left (45 \, {\left (b d^{5} x^{4} + 4 \, b c d^{4} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c^{3} d^{2} x + b c^{4} d\right )} \arcsin \left (d x + c\right ) e^{3} + 2 \, {\left (5 \, b d^{4} x^{3} + 15 \, b c d^{3} x^{2} + {\left (15 \, b c^{2} + 7 \, b\right )} d^{2} x + {\left (5 \, b c^{3} + 7 \, b c\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} e^{3} + 45 \, {\left (a d^{5} x^{4} + 4 \, a c d^{4} x^{3} + 6 \, a c^{2} d^{3} x^{2} + 4 \, a c^{3} d^{2} x + a c^{4} d\right )} e^{3}\right )} \sqrt {d x + c} e^{\frac {1}{2}}\right )}}{405 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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