\(\int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx\) [56]

Optimal result

Integrand size = 18, antiderivative size = 381 \[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=-\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {45 d^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{7/2}}+\frac {5 d^{5/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}-\frac {45 d^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{7/2}}-\frac {5 d^{5/2} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \sinh (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {5 d^2 \sqrt {c+d x} \sinh (3 a+3 b x)}{144 b^3} \]

[Out]

-5/3*d*(d*x+c)^(3/2)*cosh(b*x+a)/b^2-5/18*d*(d*x+c)^(3/2)*cosh(b*x+a)^3/b^2+2/3*(d*x+c)^(5/2)*sinh(b*x+a)/b+1/
3*(d*x+c)^(5/2)*cosh(b*x+a)^2*sinh(b*x+a)/b+5/1728*d^(5/2)*exp(-3*a+3*b*c/d)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)
/d^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)-5/1728*d^(5/2)*exp(3*a-3*b*c/d)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))
*3^(1/2)*Pi^(1/2)/b^(7/2)+45/64*d^(5/2)*exp(-a+b*c/d)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(7/2)-45/6
4*d^(5/2)*exp(a-b*c/d)*erfi(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(7/2)+45/16*d^2*sinh(b*x+a)*(d*x+c)^(1/2
)/b^3+5/144*d^2*sinh(3*b*x+3*a)*(d*x+c)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3392, 3377, 3389, 2211, 2235, 2236, 3393} \[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=\frac {45 \sqrt {\pi } d^{5/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{7/2}}+\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}-\frac {45 \sqrt {\pi } d^{5/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \sinh (a+b x)}{16 b^3}+\frac {5 d^2 \sqrt {c+d x} \sinh (3 a+3 b x)}{144 b^3}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}-\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]

[In]

Int[(c + d*x)^(5/2)*Cosh[a + b*x]^3,x]

[Out]

(-5*d*(c + d*x)^(3/2)*Cosh[a + b*x])/(3*b^2) - (5*d*(c + d*x)^(3/2)*Cosh[a + b*x]^3)/(18*b^2) + (45*d^(5/2)*E^
(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(7/2)) + (5*d^(5/2)*E^(-3*a + (3*b*c)/d)*S
qrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (45*d^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*E
rfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(7/2)) - (5*d^(5/2)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*S
qrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (45*d^2*Sqrt[c + d*x]*Sinh[a + b*x])/(16*b^3) + (2*(c + d*x)^(
5/2)*Sinh[a + b*x])/(3*b) + ((c + d*x)^(5/2)*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (5*d^2*Sqrt[c + d*x]*Sinh[
3*a + 3*b*x])/(144*b^3)

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3389

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

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rubi steps \begin{align*} \text {integral}& = -\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^{5/2} \cosh (a+b x) \, dx+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cosh ^3(a+b x) \, dx}{12 b^2} \\ & = -\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {(5 d) \int (c+d x)^{3/2} \sinh (a+b x) \, dx}{3 b}+\frac {\left (5 d^2\right ) \int \left (\frac {3}{4} \sqrt {c+d x} \cosh (a+b x)+\frac {1}{4} \sqrt {c+d x} \cosh (3 a+3 b x)\right ) \, dx}{12 b^2} \\ & = -\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cosh (3 a+3 b x) \, dx}{48 b^2}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cosh (a+b x) \, dx}{16 b^2}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cosh (a+b x) \, dx}{2 b^2} \\ & = -\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {45 d^2 \sqrt {c+d x} \sinh (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {5 d^2 \sqrt {c+d x} \sinh (3 a+3 b x)}{144 b^3}-\frac {\left (5 d^3\right ) \int \frac {\sinh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{288 b^3}-\frac {\left (5 d^3\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^3}-\frac {\left (5 d^3\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{4 b^3} \\ & = -\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {45 d^2 \sqrt {c+d x} \sinh (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {5 d^2 \sqrt {c+d x} \sinh (3 a+3 b x)}{144 b^3}-\frac {\left (5 d^3\right ) \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{576 b^3}+\frac {\left (5 d^3\right ) \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{576 b^3}-\frac {\left (5 d^3\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{64 b^3}+\frac {\left (5 d^3\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{64 b^3}-\frac {\left (5 d^3\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{8 b^3}+\frac {\left (5 d^3\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{8 b^3} \\ & = -\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {45 d^2 \sqrt {c+d x} \sinh (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {5 d^2 \sqrt {c+d x} \sinh (3 a+3 b x)}{144 b^3}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{288 b^3}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{288 b^3}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{32 b^3}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{32 b^3}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b^3}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b^3} \\ & = -\frac {5 d (c+d x)^{3/2} \cosh (a+b x)}{3 b^2}-\frac {5 d (c+d x)^{3/2} \cosh ^3(a+b x)}{18 b^2}+\frac {45 d^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{7/2}}+\frac {5 d^{5/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}-\frac {45 d^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{7/2}}-\frac {5 d^{5/2} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \sinh (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {5 d^2 \sqrt {c+d x} \sinh (3 a+3 b x)}{144 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.51 \[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=\frac {e^{-3 \left (a+\frac {b c}{d}\right )} (c+d x)^{3/2} \left (\sqrt {3} b e^{6 a} (c+d x) \Gamma \left (\frac {7}{2},-\frac {3 b (c+d x)}{d}\right )+243 b e^{4 a+\frac {2 b c}{d}} (c+d x) \Gamma \left (\frac {7}{2},-\frac {b (c+d x)}{d}\right )-d e^{\frac {4 b c}{d}} \sqrt {-\frac {b^2 (c+d x)^2}{d^2}} \left (243 e^{2 a} \Gamma \left (\frac {7}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {7}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{648 b^2 \left (-\frac {b (c+d x)}{d}\right )^{5/2}} \]

[In]

Integrate[(c + d*x)^(5/2)*Cosh[a + b*x]^3,x]

[Out]

((c + d*x)^(3/2)*(Sqrt[3]*b*E^(6*a)*(c + d*x)*Gamma[7/2, (-3*b*(c + d*x))/d] + 243*b*E^(4*a + (2*b*c)/d)*(c +
d*x)*Gamma[7/2, -((b*(c + d*x))/d)] - d*E^((4*b*c)/d)*Sqrt[-((b^2*(c + d*x)^2)/d^2)]*(243*E^(2*a)*Gamma[7/2, (
b*(c + d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[7/2, (3*b*(c + d*x))/d])))/(648*b^2*E^(3*(a + (b*c)/d))*(-((b*(c
+ d*x))/d))^(5/2))

Maple [F]

\[\int \left (d x +c \right )^{\frac {5}{2}} \cosh \left (b x +a \right )^{3}d x\]

[In]

int((d*x+c)^(5/2)*cosh(b*x+a)^3,x)

[Out]

int((d*x+c)^(5/2)*cosh(b*x+a)^3,x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2092 vs. \(2 (291) = 582\).

Time = 0.29 (sec) , antiderivative size = 2092, normalized size of antiderivative = 5.49 \[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(5/2)*cosh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/1728*(5*sqrt(3)*sqrt(pi)*(d^3*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - d^3*cosh(b*x + a)^3*sinh(-3*(b*c - a*
d)/d) + (d^3*cosh(-3*(b*c - a*d)/d) - d^3*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^3*cosh(b*x + a)*cosh(
-3*(b*c - a*d)/d) - d^3*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^3*cosh(b*x + a)^2*cosh(-3
*(b*c - a*d)/d) - d^3*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(3)*sqrt(d*x +
c)*sqrt(b/d)) + 5*sqrt(3)*sqrt(pi)*(d^3*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) + d^3*cosh(b*x + a)^3*sinh(-3*(
b*c - a*d)/d) + (d^3*cosh(-3*(b*c - a*d)/d) + d^3*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^3*cosh(b*x +
a)*cosh(-3*(b*c - a*d)/d) + d^3*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^3*cosh(b*x + a)^2
*cosh(-3*(b*c - a*d)/d) + d^3*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(3)*sq
rt(d*x + c)*sqrt(-b/d)) + 1215*sqrt(pi)*(d^3*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) - d^3*cosh(b*x + a)^3*sinh(-
(b*c - a*d)/d) + (d^3*cosh(-(b*c - a*d)/d) - d^3*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^3*cosh(b*x + a)*
cosh(-(b*c - a*d)/d) - d^3*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^3*cosh(b*x + a)^2*cosh(-
(b*c - a*d)/d) - d^3*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d
)) + 1215*sqrt(pi)*(d^3*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + d^3*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + (d^3
*cosh(-(b*c - a*d)/d) + d^3*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^3*cosh(b*x + a)*cosh(-(b*c - a*d)/d)
+ d^3*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^3*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) + d^3*
cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) - 6*(12*b^3*d^2*
x^2 - (12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*b*d^2 + 2*(12*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)^6 - 6*
(12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*b*d^2 + 2*(12*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)*sinh(b*x + a
)^5 - (12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*b*d^2 + 2*(12*b^3*c*d - 5*b^2*d^2)*x)*sinh(b*x + a)^6 + 12
*b^3*c^2 - 27*(4*b^3*d^2*x^2 + 4*b^3*c^2 - 10*b^2*c*d + 15*b*d^2 + 2*(4*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)^
4 - 3*(36*b^3*d^2*x^2 + 36*b^3*c^2 - 90*b^2*c*d + 135*b*d^2 + 5*(12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*
b*d^2 + 2*(12*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)^2 + 18*(4*b^3*c*d - 5*b^2*d^2)*x)*sinh(b*x + a)^4 + 10*b^2
*c*d - 4*(5*(12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*b*d^2 + 2*(12*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)^
3 + 27*(4*b^3*d^2*x^2 + 4*b^3*c^2 - 10*b^2*c*d + 15*b*d^2 + 2*(4*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a))*sinh(b
*x + a)^3 + 5*b*d^2 + 27*(4*b^3*d^2*x^2 + 4*b^3*c^2 + 10*b^2*c*d + 15*b*d^2 + 2*(4*b^3*c*d + 5*b^2*d^2)*x)*cos
h(b*x + a)^2 + 3*(36*b^3*d^2*x^2 + 36*b^3*c^2 - 5*(12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*b*d^2 + 2*(12*
b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)^4 + 90*b^2*c*d + 135*b*d^2 - 54*(4*b^3*d^2*x^2 + 4*b^3*c^2 - 10*b^2*c*d
+ 15*b*d^2 + 2*(4*b^3*c*d - 5*b^2*d^2)*x)*cosh(b*x + a)^2 + 18*(4*b^3*c*d + 5*b^2*d^2)*x)*sinh(b*x + a)^2 + 2*
(12*b^3*c*d + 5*b^2*d^2)*x - 6*((12*b^3*d^2*x^2 + 12*b^3*c^2 - 10*b^2*c*d + 5*b*d^2 + 2*(12*b^3*c*d - 5*b^2*d^
2)*x)*cosh(b*x + a)^5 + 18*(4*b^3*d^2*x^2 + 4*b^3*c^2 - 10*b^2*c*d + 15*b*d^2 + 2*(4*b^3*c*d - 5*b^2*d^2)*x)*c
osh(b*x + a)^3 - 9*(4*b^3*d^2*x^2 + 4*b^3*c^2 + 10*b^2*c*d + 15*b*d^2 + 2*(4*b^3*c*d + 5*b^2*d^2)*x)*cosh(b*x
+ a))*sinh(b*x + a))*sqrt(d*x + c))/(b^4*cosh(b*x + a)^3 + 3*b^4*cosh(b*x + a)^2*sinh(b*x + a) + 3*b^4*cosh(b*
x + a)*sinh(b*x + a)^2 + b^4*sinh(b*x + a)^3)

Sympy [F]

\[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {5}{2}} \cosh ^{3}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**(5/2)*cosh(b*x+a)**3,x)

[Out]

Integral((c + d*x)**(5/2)*cosh(a + b*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.35 \[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=-\frac {\frac {5 \, \sqrt {3} \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b^{3} \sqrt {-\frac {b}{d}}} - \frac {5 \, \sqrt {3} \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b^{3} \sqrt {\frac {b}{d}}} + \frac {1215 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{3} \sqrt {-\frac {b}{d}}} - \frac {1215 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{3} \sqrt {\frac {b}{d}}} + \frac {162 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (\frac {b c}{d}\right )} + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (\frac {b c}{d}\right )} + 15 \, \sqrt {d x + c} d^{3} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{3}} + \frac {6 \, {\left (12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (\frac {3 \, b c}{d}\right )} + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (\frac {3 \, b c}{d}\right )} + 5 \, \sqrt {d x + c} d^{3} e^{\left (\frac {3 \, b c}{d}\right )}\right )} e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d}\right )}}{b^{3}} - \frac {6 \, {\left (12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (3 \, a\right )} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (3 \, a\right )} + 5 \, \sqrt {d x + c} d^{3} e^{\left (3 \, a\right )}\right )} e^{\left (\frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b^{3}} - \frac {162 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{a} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{a} + 15 \, \sqrt {d x + c} d^{3} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{3}}}{1728 \, d} \]

[In]

integrate((d*x+c)^(5/2)*cosh(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/1728*(5*sqrt(3)*sqrt(pi)*d^3*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 3*b*c/d)/(b^3*sqrt(-b/d)) - 5*s
qrt(3)*sqrt(pi)*d^3*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d))*e^(-3*a + 3*b*c/d)/(b^3*sqrt(b/d)) + 1215*sqrt(pi)*d^
3*erf(sqrt(d*x + c)*sqrt(-b/d))*e^(a - b*c/d)/(b^3*sqrt(-b/d)) - 1215*sqrt(pi)*d^3*erf(sqrt(d*x + c)*sqrt(b/d)
)*e^(-a + b*c/d)/(b^3*sqrt(b/d)) + 162*(4*(d*x + c)^(5/2)*b^2*d*e^(b*c/d) + 10*(d*x + c)^(3/2)*b*d^2*e^(b*c/d)
+ 15*sqrt(d*x + c)*d^3*e^(b*c/d))*e^(-a - (d*x + c)*b/d)/b^3 + 6*(12*(d*x + c)^(5/2)*b^2*d*e^(3*b*c/d) + 10*(
d*x + c)^(3/2)*b*d^2*e^(3*b*c/d) + 5*sqrt(d*x + c)*d^3*e^(3*b*c/d))*e^(-3*a - 3*(d*x + c)*b/d)/b^3 - 6*(12*(d*
x + c)^(5/2)*b^2*d*e^(3*a) - 10*(d*x + c)^(3/2)*b*d^2*e^(3*a) + 5*sqrt(d*x + c)*d^3*e^(3*a))*e^(3*(d*x + c)*b/
d - 3*b*c/d)/b^3 - 162*(4*(d*x + c)^(5/2)*b^2*d*e^a - 10*(d*x + c)^(3/2)*b*d^2*e^a + 15*sqrt(d*x + c)*d^3*e^a)
*e^((d*x + c)*b/d - b*c/d)/b^3)/d

Giac [F]

\[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{\frac {5}{2}} \cosh \left (b x + a\right )^{3} \,d x } \]

[In]

integrate((d*x+c)^(5/2)*cosh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/2)*cosh(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{5/2} \cosh ^3(a+b x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \]

[In]

int(cosh(a + b*x)^3*(c + d*x)^(5/2),x)

[Out]

int(cosh(a + b*x)^3*(c + d*x)^(5/2), x)