Optimal. Leaf size=542 \[ \frac {d 5^{-n-1} e^{-\frac {5 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 3^{-n} e^{-\frac {3 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d e^{-\frac {a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {c^2 x^2+1}}+\frac {d e^{a/b} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {c^2 x^2+1}}+\frac {d 3^{-n} e^{\frac {3 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 5^{-n-1} e^{\frac {5 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.63, antiderivative size = 542, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5782, 5779, 5448, 3308, 2181} \[ \frac {d 5^{-n-1} e^{-\frac {5 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 3^{-n} e^{-\frac {3 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d e^{-\frac {a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {c^2 x^2+1}}+\frac {d e^{a/b} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {c^2 x^2+1}}+\frac {d 3^{-n} e^{\frac {3 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 5^{-n-1} e^{\frac {5 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3308
Rule 5448
Rule 5779
Rule 5782
Rubi steps
\begin {align*} \int x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh ^4(x) \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{8} (a+b x)^n \sinh (x)+\frac {3}{16} (a+b x)^n \sinh (3 x)+\frac {1}{16} (a+b x)^n \sinh (5 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sinh (5 x) \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \sinh (3 x) \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {5^{-1-n} d e^{-\frac {5 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {d e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {d e^{a/b} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-1-n} d e^{\frac {5 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.79, size = 390, normalized size = 0.72 \[ \frac {d^2 15^{-n-1} e^{-\frac {5 a}{b}} \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \left (3^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+5^{n+1} e^{\frac {2 a}{b}} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2\ 3^n 5^{n+1} e^{\frac {4 a}{b}} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )+5^{n+1} e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \Gamma \left (n+1,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3^n e^{\frac {10 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \Gamma \left (n+1,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )+2\ 15^{n+1} e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{32 c^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{2} d x^{3} + d x\right )} \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________