Optimal. Leaf size=99 \[ \frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {ArcSin}(c+d x))}{3 d e}-\frac {4 b \sqrt {e} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4889, 4723,
327, 335, 227} \begin {gather*} \frac {2 (e (c+d x))^{3/2} (a+b \text {ArcSin}(c+d x))}{3 d e}-\frac {4 b \sqrt {e} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d}+\frac {4 b \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}}{9 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 327
Rule 335
Rule 4723
Rule 4889
Rubi steps
\begin {align*} \int \sqrt {c e+d e x} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \sqrt {e x} \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e}\\ &=\frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b e) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=\frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{9 d}\\ &=\frac {4 b \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {e} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.02, size = 87, normalized size = 0.88 \begin {gather*} \frac {2 \sqrt {e (c+d x)} \left (3 a c+3 a d x+2 b \sqrt {1-(c+d x)^2}+3 b c \text {ArcSin}(c+d x)+3 b d x \text {ArcSin}(c+d x)-2 b \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )\right )}{9 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 171 vs. \(2 (81 ) = 162\).
time = 0.14, size = 172, normalized size = 1.74
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (-\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}+\frac {e^{2} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{3 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\) | \(172\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (-\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}+\frac {e^{2} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{3 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\) | \(172\) |
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 = 0.89, size = 108, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left ({\left (3 \, a d^{3} x + 3 \, a c d^{2} + 2 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} b d^{2} + 3 \, {\left (b d^{3} x + b c d^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {d x + c} e^{\frac {1}{2}} + 2 \, \sqrt {-d^{3} e} b {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{9 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.34, size = 104, normalized size = 1.05 \begin {gather*} \frac {2 a \left (c e + d e x\right )^{\frac {3}{2}}}{3 d e} + \frac {2 b \left (c e + d e x\right )^{\frac {3}{2}} \operatorname {asin}{\left (c + d x \right )}}{3 d e} - \frac {b \left (c e + d e x\right )^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {\left (c e + d e x\right )^{2} e^{2 i \pi }}{e^{2}}} \right )}}{3 d e^{2} \Gamma \left (\frac {9}{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 \sqrt {c\,e+d\,e\,x}\,\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]
________________________________________________________________________________________