Optimal. Leaf size=117 \[ \frac {4 b (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{25 d}+\frac {2 (e (c+d x))^{5/2} (a+b \text {ArcSin}(c+d x))}{5 d e}+\frac {12 b e \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, 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 = {4889, 4723,
327, 326, 324, 435} \begin {gather*} \frac {2 (e (c+d x))^{5/2} (a+b \text {ArcSin}(c+d x))}{5 d e}+\frac {12 b e \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {c+d x}}+\frac {4 b \sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}}{25 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 324
Rule 326
Rule 327
Rule 435
Rule 4723
Rule 4889
Rubi steps
\begin {align*} \int (c e+d e x)^{3/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\text {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) \text {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) \text {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 ) \text {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 ) \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 )}{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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.03, size = 87, normalized size = 0.74 \begin {gather*} \frac {2 (e (c+d x))^{3/2} \left (5 a c+5 a d x+2 b \sqrt {1-(c+d x)^2}+5 b c \text {ArcSin}(c+d x)+5 b d x \text {ArcSin}(c+d x)-2 b \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )\right )}{25 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.13, size = 194, normalized size = 1.66
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{5}-\frac {3 e^{3} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{5 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{5 e}\right )}{d e}\) | \(194\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{5}-\frac {3 e^{3} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{5 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{5 e}\right )}{d e}\) | \(194\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.56, size = 152, normalized size = 1.30 \begin {gather*} -\frac {2 \, {\left (6 \, \sqrt {-d^{3} e} b e {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - {\left (5 \, {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \arcsin \left (d x + c\right ) e + 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} e + 5 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )} e\right )} \sqrt {d x + c} e^{\frac {1}{2}}\right )}}{25 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 10.80, size = 156, normalized size = 1.33 \begin {gather*} 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 )} \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 {\left (c\,e+d\,e\,x\right )}^{3/2}\,\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.
[In]
[Out]
________________________________________________________________________________________