Optimal. Leaf size=130 \[ \frac {2 (e (c+d x))^{3/2} (a+b \text {ArcSin}(c+d x))^2}{3 d e}-\frac {8 b (e (c+d x))^{5/2} (a+b \text {ArcSin}(c+d x)) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};(c+d x)^2\right )}{15 d e^2}+\frac {16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};(c+d x)^2\right )}{105 d e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, 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 = {4889, 4723,
4805} \begin {gather*} \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 ) (a+b \text {ArcSin}(c+d x))}{15 d e^2}+\frac {2 (e (c+d x))^{3/2} (a+b \text {ArcSin}(c+d x))^2}{3 d e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4723
Rule 4805
Rule 4889
Rubi steps
\begin {align*} \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {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) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 107, normalized size = 0.82 \begin {gather*} \frac {2 (e (c+d x))^{3/2} \left (7 (a+b \text {ArcSin}(c+d x)) \left (5 (a+b \text {ArcSin}(c+d x))-4 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};(c+d x)^2\right )\right )+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 )\right )}{105 d e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.20, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________