Optimal. Leaf size=175 \[ \frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{2 d}+\frac {(c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4887, 4715,
4767, 4719, 3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{2 d}+\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{2 d}+\frac {(c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4715
Rule 4719
Rule 4767
Rule 4887
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2 d}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2 d}\\ &=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 2.00, size = 313, normalized size = 1.79 \begin {gather*} \frac {b \left (2 \sqrt {a+b \text {ArcSin}(c+d x)} \left (3 \sqrt {1-(c+d x)^2}+2 (c+d x) \text {ArcSin}(c+d x)\right )+\frac {2 a e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{\sqrt {a+b \text {ArcSin}(c+d x)}}-\sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs.
\(2(139)=278\).
time = 0.27, size = 304, normalized size = 1.74
method | result | size |
default | \(-\frac {3 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-3 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}+4 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}-6 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b -6 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b +4 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2}}{4 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(304\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.89, size = 1061, normalized size = 6.06 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } a^{2} b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} d} + \frac {i \, \sqrt {2} \sqrt {\pi } a b^{3} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {2} \sqrt {\pi } a^{2} b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {2} \sqrt {\pi } a b^{3} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {2} \sqrt {\pi } a b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {3 \, \sqrt {2} \sqrt {\pi } b^{3} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{8 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {i \, \sqrt {2} \sqrt {\pi } a b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {3 \, \sqrt {2} \sqrt {\pi } b^{3} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{8 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } a^{2} b \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{{\left (\frac {i \, \sqrt {2} b^{2}}{\sqrt {{\left | b \right |}}} + \sqrt {2} b \sqrt {{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } a^{2} b \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{{\left (-\frac {i \, \sqrt {2} b^{2}}{\sqrt {{\left | b \right |}}} + \sqrt {2} b \sqrt {{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} b \arcsin \left (d x + c\right ) e^{\left (i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} b \arcsin \left (d x + c\right ) e^{\left (-i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} a e^{\left (i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} + \frac {3 \, \sqrt {b \arcsin \left (d x + c\right ) + a} b e^{\left (i \, \arcsin \left (d x + c\right )\right )}}{4 \, d} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} a e^{\left (-i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} + \frac {3 \, \sqrt {b \arcsin \left (d x + c\right ) + a} b e^{\left (-i \, \arcsin \left (d x + c\right )\right )}}{4 \, d} \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 (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________