\(\int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx\) [148]

Optimal result

Integrand size = 16, antiderivative size = 337 \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {5 b^2 x \sqrt {a+b \text {arccosh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{576 c^3} \] Output:

5/6*b^2*x*(a+b*arccosh(c*x))^(1/2)/c^2+5/36*b^2*x^3*(a+b*arccosh(c*x))^(1/ 
2)-5/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(3/2)/c^3-5/18*b*x 
^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(3/2)/c+1/3*x^3*(a+b*arc 
cosh(c*x))^(5/2)-15/64*b^(5/2)*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1 
/2)/b^(1/2))/c^3-5/1728*b^(5/2)*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a 
+b*arccosh(c*x))^(1/2)/b^(1/2))/c^3-15/64*b^(5/2)*Pi^(1/2)*erfi((a+b*arcco 
sh(c*x))^(1/2)/b^(1/2))/c^3/exp(a/b)-5/1728*b^(5/2)*3^(1/2)*Pi^(1/2)*erfi( 
3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/c^3/exp(3*a/b)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(924\) vs. \(2(337)=674\).

Time = 7.61 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.74 \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx =\text {Too large to display} \] Input:

Integrate[x^2*(a + b*ArcCosh[c*x])^(5/2),x]
 

Output:

(a^2*Sqrt[a + b*ArcCosh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b 
)]*Gamma[3/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma 
[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x] 
]*Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + 
b*ArcCosh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3 
*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]) + (a*Sqrt[b]*(9*(-12*Sqrt[b]*S 
qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*Sqrt[b]*c 
*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[ 
a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi 
]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + 
b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a) 
/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCos 
h[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b* 
ArcCosh[c*x]]*(2*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]] - Sinh[3*ArcCosh[c*x]]) 
))/(144*c^3) - (27*(-4*b*Sqrt[a + b*ArcCosh[c*x]]*(2*Sqrt[(-1 + c*x)/(1 + 
c*x)]*(1 + c*x)*(a - 5*b*ArcCosh[c*x]) + b*c*x*(15 + 4*ArcCosh[c*x]^2)) + 
Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/S 
qrt[b]]*(Cosh[a/b] - Sinh[a/b]) + Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[P 
i]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b 
]*(12*a^2 + 12*a*b + 5*b^2)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh...
 

Rubi [A] (verified)

Time = 5.02 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.30, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6299, 6354, 6299, 6330, 6294, 6368, 3042, 3788, 26, 2611, 2633, 2634, 3793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx\)

\(\Big \downarrow \) 6299

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \int \frac {x^3 (a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}-\frac {b \int x^2 \sqrt {a+b \text {arccosh}(c x)}dx}{2 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 6299

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\right )}{2 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \int \sqrt {a+b \text {arccosh}(c x)}dx}{2 c}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\right )}{2 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 6294

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} b c \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\right )}{2 c}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\right )}{2 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {1}{2} i \int -\frac {i e^{-\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i \int \frac {i e^{\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {1}{2} \int \frac {e^{-\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))+\frac {1}{2} \int \frac {e^{\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}+\int e^{\frac {a+b \text {arccosh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c x)}}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{3 c^2}\right )\)

Input:

Int[x^2*(a + b*ArcCosh[c*x])^(5/2),x]
 

Output:

(x^3*(a + b*ArcCosh[c*x])^(5/2))/3 - (5*b*c*((x^2*Sqrt[-1 + c*x]*Sqrt[1 + 
c*x]*(a + b*ArcCosh[c*x])^(3/2))/(3*c^2) + (2*((Sqrt[-1 + c*x]*Sqrt[1 + c* 
x]*(a + b*ArcCosh[c*x])^(3/2))/c^2 - (3*b*(x*Sqrt[a + b*ArcCosh[c*x]] - (( 
Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/2 + (Sqrt[ 
b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*E^(a/b)))/(2*c)))/( 
2*c)))/(3*c^2) - (b*((x^3*Sqrt[a + b*ArcCosh[c*x]])/3 - ((3*Sqrt[b]*E^(a/b 
)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/8 + (Sqrt[b]*E^((3*a)/b) 
*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/8 + (3*Sqrt[b 
]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]* 
Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*E^((3*a)/b 
)))/(6*c^3)))/(2*c)))/6
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6299

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1))   Int 
[x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x 
], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
 

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 
 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {5}{2}}d x\]

Input:

int(x^2*(a+b*arccosh(c*x))^(5/2),x)
 

Output:

int(x^2*(a+b*arccosh(c*x))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2*(a+b*arccosh(c*x))^(5/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+b*acosh(c*x))**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arccosh(c*x))^(5/2),x, algorithm="maxima")
 

Output:

integrate((b*arccosh(c*x) + a)^(5/2)*x^2, x)
 

Giac [F(-2)]

Exception generated. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^2*(a+b*arccosh(c*x))^(5/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2} \,d x \] Input:

int(x^2*(a + b*acosh(c*x))^(5/2),x)
 

Output:

int(x^2*(a + b*acosh(c*x))^(5/2), x)
 

Reduce [F]

\[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=2 \left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) a b +\left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2}+\left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x^{2}d x \right ) a^{2} \] Input:

int(x^2*(a+b*acosh(c*x))^(5/2),x)
 

Output:

2*int(sqrt(acosh(c*x)*b + a)*acosh(c*x)*x**2,x)*a*b + int(sqrt(acosh(c*x)* 
b + a)*acosh(c*x)**2*x**2,x)*b**2 + int(sqrt(acosh(c*x)*b + a)*x**2,x)*a** 
2