\(\int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) [84]

Optimal result

Integrand size = 25, antiderivative size = 408 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \text {arccosh}(c+d x)}}{36 d}-\frac {5 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{9 d}-\frac {5 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^{5/2}}{3 d}-\frac {15 b^{5/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 b^{5/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{576 d}-\frac {15 b^{5/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 b^{5/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{576 d} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1008\) vs. \(2(408)=816\).

Time = 8.70 (sec) , antiderivative size = 1008, normalized size of antiderivative = 2.47 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

e^2*((a^2*Sqrt[a + b*ArcCosh[c + d*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCos 
h[c + d*x])/b)]*Gamma[3/2, a/b + ArcCosh[c + d*x]] + Sqrt[3]*Sqrt[a/b + Ar 
cCosh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 9*E^((2*a)/b 
)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)] + 
 Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[3/2, (3*(a 
+ b*ArcCosh[c + d*x]))/b]))/(72*d*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c + d* 
x])^2/b^2)]) + (a*Sqrt[b]*(9*(-12*Sqrt[b]*Sqrt[(-1 + c + d*x)/(1 + c + d*x 
)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*Sqrt[b]*(c + d*x)*ArcCos 
h[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a 
 + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt 
[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + 
(2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]* 
(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a 
 + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt 
[b]*Sqrt[a + b*ArcCosh[c + d*x]]*(2*ArcCosh[c + d*x]*Cosh[3*ArcCosh[c + d* 
x]] - Sinh[3*ArcCosh[c + d*x]])))/(144*d) + (-27*(-4*b*Sqrt[a + b*ArcCosh[ 
c + d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*Arc 
Cosh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcCosh[c + d*x]^2)) + Sqrt[b]*(4*a^2 
 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(C 
osh[a/b] - Sinh[a/b]) + Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[...
 

Rubi [A] (verified)

Time = 4.66 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {6411, 27, 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.

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

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^2*(((c + d*x)^3*(a + b*ArcCosh[c + d*x])^(5/2))/3 - (5*b*((Sqrt[-1 + c 
+ d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/3 + ( 
2*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2) - ( 
3*b*((c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + (-1/2*(Sqrt[b]*E^(a/b)*Sqrt[ 
Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sq 
rt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(2*E^(a/b)))/2))/2))/3 - (b*(((c + d* 
x)^3*Sqrt[a + b*ArcCosh[c + d*x]])/3 + ((-3*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[S 
qrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/8 - (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]* 
Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/8 - (3*Sqrt[b]*Sqrt[P 
i]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) - (Sqrt[b]*Sqrt 
[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b 
)))/6))/2))/6))/d
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 6411

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^{5/2} \, dx=e^{2} \left (\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}d x \right ) a^{2} c^{2}+2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right ) x^{2}d x \right ) a b \,d^{2}+4 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right ) x d x \right ) a b c d +2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )d x \right ) a b \,c^{2}+\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} d^{2}+2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} c d +\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c^{2}+\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, x^{2}d x \right ) a^{2} d^{2}+2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, x d x \right ) a^{2} c d \right ) \] Input:

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

Output:

e**2*(int(sqrt(acosh(c + d*x)*b + a),x)*a**2*c**2 + 2*int(sqrt(acosh(c + d 
*x)*b + a)*acosh(c + d*x)*x**2,x)*a*b*d**2 + 4*int(sqrt(acosh(c + d*x)*b + 
 a)*acosh(c + d*x)*x,x)*a*b*c*d + 2*int(sqrt(acosh(c + d*x)*b + a)*acosh(c 
 + d*x),x)*a*b*c**2 + int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2*x** 
2,x)*b**2*d**2 + 2*int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2*x,x)*b 
**2*c*d + int(sqrt(acosh(c + d*x)*b + a)*acosh(c + d*x)**2,x)*b**2*c**2 + 
int(sqrt(acosh(c + d*x)*b + a)*x**2,x)*a**2*d**2 + 2*int(sqrt(acosh(c + d* 
x)*b + a)*x,x)*a**2*c*d)