Optimal. Leaf size=269 \[ \frac {\sqrt {\pi } c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {\sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {2 c \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.54, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5865, 5803, 5655, 5779, 3308, 2180, 2204, 2205, 5665, 3307} \[ \frac {\sqrt {\pi } c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {\sqrt {\pi } c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {2 c \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3308
Rule 5655
Rule 5665
Rule 5779
Rule 5803
Rule 5865
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {c}{d \left (a+b \sinh ^{-1}(x)\right )^{3/2}}+\frac {x}{d \left (a+b \sinh ^{-1}(x)\right )^{3/2}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}+\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {2 \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b^2 d^2}+\frac {2 \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b^2 d^2}+\frac {c \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}-\frac {c \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {(2 c) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b^2 d^2}-\frac {(2 c) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b^2 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.69, size = 301, normalized size = 1.12 \[ \frac {\sqrt {2 \pi } \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\frac {2 \sqrt {b} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{\sqrt {a+b \sinh ^{-1}(c+d x)}}}{2 b^{3/2} d^2}-\frac {c e^{-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \left (-e^{a/b} \left (e^{2 \sinh ^{-1}(c+d x)}+1\right )+e^{\frac {2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+e^{\sinh ^{-1}(c+d x)} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )\right )}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________