Optimal. Leaf size=616 \[ -\frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{16 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {d 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}+\frac {d 2^{-2 n-7} e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-n-7} e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}+\frac {d 2^{-n-7} e^{\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-2 n-7} e^{\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.83, antiderivative size = 616, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5782, 5779, 5448, 3307, 2181} \[ \frac {d 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}+\frac {d 2^{-2 n-7} e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-n-7} e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}+\frac {d 2^{-n-7} e^{\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-2 n-7} e^{\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{16 b c^3 (n+1) \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3307
Rule 5448
Rule 5779
Rule 5782
Rubi steps
\begin {align*} \int x^2 \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^2 \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 ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{16} (a+b x)^n-\frac {1}{32} (a+b x)^n \cosh (2 x)+\frac {1}{16} (a+b x)^n \cosh (4 x)+\frac {1}{32} (a+b x)^n \cosh (6 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{16 b c^3 (1+n) \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 (2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \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 (6 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \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) \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-6 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{6 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d e^{-\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-2 n} d e^{-\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d e^{-\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d e^{\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-2 n} d e^{\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d e^{\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.19, size = 429, normalized size = 0.70 \[ -\frac {d^2 2^{-2 n-7} 3^{-n-1} e^{-\frac {6 a}{b}} \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (2^n e^{\frac {6 a}{b}} \left (2^{n+3} 3^{n+1} \left (a+b \sinh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n+b (n+1) e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )+b \left (-2^n\right ) (n+1) \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b 3^{n+1} (n+1) e^{\frac {2 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 2^n 3^{n+1} (n+1) e^{\frac {4 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b 2^n 3^{n+1} (n+1) e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} (n+1) e^{\frac {10 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (n+1) \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{2} d x^{4} + d x^{2}\right )} \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
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]
________________________________________________________________________________________