\(\int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx\) [54]

Optimal result

Integrand size = 18, antiderivative size = 325 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=-\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {9 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}-\frac {d^{3/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {9 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2} \]

[Out]

-2/3*(d*x+c)^(3/2)*cosh(b*x+a)/b+1/3*(d*x+c)^(3/2)*cosh(b*x+a)*sinh(b*x+a)^2/b-1/288*d^(3/2)*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^(5/2)+1/288*d^(3/2)*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^(5/2)+9/32*d^(3/2)*exp(-a+b*c/d)*erf(b^(1/2)*(d*x+c)^(1/2
)/d^(1/2))*Pi^(1/2)/b^(5/2)-9/32*d^(3/2)*exp(a-b*c/d)*erfi(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(5/2)+d*s
inh(b*x+a)*(d*x+c)^(1/2)/b^2-1/6*d*sinh(b*x+a)^3*(d*x+c)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3392, 3377, 3389, 2211, 2235, 2236, 3393} \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\frac {9 \sqrt {\pi } d^{3/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}-\frac {\sqrt {\frac {\pi }{3}} d^{3/2} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}-\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]

[In]

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

[Out]

(-2*(c + d*x)^(3/2)*Cosh[a + b*x])/(3*b) + (9*d^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sq
rt[d]])/(32*b^(5/2)) - (d^(3/2)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/
(96*b^(5/2)) - (9*d^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(5/2)) + (d^(3
/2)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (d*Sqrt[c + d
*x]*Sinh[a + b*x])/b^2 + ((c + d*x)^(3/2)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (d*Sqrt[c + d*x]*Sinh[a + b*x
]^3)/(6*b^2)

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 {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {2}{3} \int (c+d x)^{3/2} \sinh (a+b x) \, dx+\frac {d^2 \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx}{12 b^2} \\ & = -\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {d \int \sqrt {c+d x} \cosh (a+b x) \, dx}{b}+\frac {\left (i d^2\right ) \int \left (\frac {3 i \sinh (a+b x)}{4 \sqrt {c+d x}}-\frac {i \sinh (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{12 b^2} \\ & = -\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {d^2 \int \frac {\sinh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{48 b^2}-\frac {d^2 \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{16 b^2}-\frac {d^2 \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{2 b^2} \\ & = -\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {d^2 \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{96 b^2}-\frac {d^2 \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{96 b^2}-\frac {d^2 \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{32 b^2}+\frac {d^2 \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{32 b^2}-\frac {d^2 \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 b^2}+\frac {d^2 \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 b^2} \\ & = -\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {d \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 )}{48 b^2}+\frac {d \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 )}{48 b^2}+\frac {d \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 )}{16 b^2}-\frac {d \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 )}{16 b^2}+\frac {d \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 )}{2 b^2}-\frac {d \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 )}{2 b^2} \\ & = -\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {9 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}-\frac {d^{3/2} 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 )}{96 b^{5/2}}-\frac {9 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} 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 )}{96 b^{5/2}}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.65 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\frac {d e^{-3 \left (a+\frac {b c}{d}\right )} \sqrt {c+d x} \left (-\sqrt {3} e^{6 a} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {3 b (c+d x)}{d}\right )+81 e^{4 a+\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \left (-81 e^{2 a} \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{216 b^2 \sqrt {-\frac {b^2 (c+d x)^2}{d^2}}} \]

[In]

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

[Out]

(d*Sqrt[c + d*x]*(-(Sqrt[3]*E^(6*a)*Sqrt[(b*(c + d*x))/d]*Gamma[5/2, (-3*b*(c + d*x))/d]) + 81*E^(4*a + (2*b*c
)/d)*Sqrt[(b*(c + d*x))/d]*Gamma[5/2, -((b*(c + d*x))/d)] + E^((4*b*c)/d)*Sqrt[-((b*(c + d*x))/d)]*(-81*E^(2*a
)*Gamma[5/2, (b*(c + d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[5/2, (3*b*(c + d*x))/d])))/(216*b^2*E^(3*(a + (b*c
)/d))*Sqrt[-((b^2*(c + d*x)^2)/d^2)])

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1543 vs. \(2 (245) = 490\).

Time = 0.27 (sec) , antiderivative size = 1543, normalized size of antiderivative = 4.75 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/288*(sqrt(3)*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - d^2*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)
/d) + (d^2*cosh(-3*(b*c - a*d)/d) - d^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^2*cosh(b*x + a)*cosh(-3
*(b*c - a*d)/d) - d^2*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cosh(b*x + a)^2*cosh(-3*(
b*c - a*d)/d) - d^2*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)) + sqrt(3)*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) + d^2*cosh(b*x + a)^3*sinh(-3*(b*c
- a*d)/d) + (d^2*cosh(-3*(b*c - a*d)/d) + d^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^2*cosh(b*x + a)*c
osh(-3*(b*c - a*d)/d) + d^2*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cosh(b*x + a)^2*cos
h(-3*(b*c - a*d)/d) + d^2*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)) - 81*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) - d^2*cosh(b*x + a)^3*sinh(-(b*c -
a*d)/d) + (d^2*cosh(-(b*c - a*d)/d) - d^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^2*cosh(b*x + a)*cosh(-
(b*c - a*d)/d) - d^2*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cosh(b*x + a)^2*cosh(-(b*c -
a*d)/d) - d^2*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)) - 8
1*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + d^2*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + (d^2*cosh(-(
b*c - a*d)/d) + d^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^2*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + d^2*co
sh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) + d^2*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*((2*b^2*d*x + 2*b^2*
c - b*d)*cosh(b*x + a)^6 + 6*(2*b^2*d*x + 2*b^2*c - b*d)*cosh(b*x + a)*sinh(b*x + a)^5 + (2*b^2*d*x + 2*b^2*c
- b*d)*sinh(b*x + a)^6 - 9*(2*b^2*d*x + 2*b^2*c - 3*b*d)*cosh(b*x + a)^4 - 3*(6*b^2*d*x + 6*b^2*c - 5*(2*b^2*d
*x + 2*b^2*c - b*d)*cosh(b*x + a)^2 - 9*b*d)*sinh(b*x + a)^4 + 2*b^2*d*x + 4*(5*(2*b^2*d*x + 2*b^2*c - b*d)*co
sh(b*x + a)^3 - 9*(2*b^2*d*x + 2*b^2*c - 3*b*d)*cosh(b*x + a))*sinh(b*x + a)^3 + 2*b^2*c - 9*(2*b^2*d*x + 2*b^
2*c + 3*b*d)*cosh(b*x + a)^2 + 3*(5*(2*b^2*d*x + 2*b^2*c - b*d)*cosh(b*x + a)^4 - 6*b^2*d*x - 6*b^2*c - 18*(2*
b^2*d*x + 2*b^2*c - 3*b*d)*cosh(b*x + a)^2 - 9*b*d)*sinh(b*x + a)^2 + b*d + 6*((2*b^2*d*x + 2*b^2*c - b*d)*cos
h(b*x + a)^5 - 6*(2*b^2*d*x + 2*b^2*c - 3*b*d)*cosh(b*x + a)^3 - 3*(2*b^2*d*x + 2*b^2*c + 3*b*d)*cosh(b*x + a)
)*sinh(b*x + a))*sqrt(d*x + c))/(b^3*cosh(b*x + a)^3 + 3*b^3*cosh(b*x + a)^2*sinh(b*x + a) + 3*b^3*cosh(b*x +
a)*sinh(b*x + a)^2 + b^3*sinh(b*x + a)^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.32 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\frac {\frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} - \frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {3 \, b c}{d}\right )} + \sqrt {d x + c} d^{2} e^{\left (\frac {3 \, b c}{d}\right )}\right )} e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (3 \, a\right )} - \sqrt {d x + c} d^{2} e^{\left (3 \, a\right )}\right )} e^{\left (\frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b^{2}} - \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{a} - 3 \, \sqrt {d x + c} d^{2} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{2}}}{288 \, d} \]

[In]

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

[Out]

1/288*(sqrt(3)*sqrt(pi)*d^2*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 3*b*c/d)/(b^2*sqrt(-b/d)) - sqrt(3)
*sqrt(pi)*d^2*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d))*e^(-3*a + 3*b*c/d)/(b^2*sqrt(b/d)) - 81*sqrt(pi)*d^2*erf(sq
rt(d*x + c)*sqrt(-b/d))*e^(a - b*c/d)/(b^2*sqrt(-b/d)) + 81*sqrt(pi)*d^2*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-a +
b*c/d)/(b^2*sqrt(b/d)) - 54*(2*(d*x + c)^(3/2)*b*d*e^(b*c/d) + 3*sqrt(d*x + c)*d^2*e^(b*c/d))*e^(-a - (d*x + c
)*b/d)/b^2 + 6*(2*(d*x + c)^(3/2)*b*d*e^(3*b*c/d) + sqrt(d*x + c)*d^2*e^(3*b*c/d))*e^(-3*a - 3*(d*x + c)*b/d)/
b^2 + 6*(2*(d*x + c)^(3/2)*b*d*e^(3*a) - sqrt(d*x + c)*d^2*e^(3*a))*e^(3*(d*x + c)*b/d - 3*b*c/d)/b^2 - 54*(2*
(d*x + c)^(3/2)*b*d*e^a - 3*sqrt(d*x + c)*d^2*e^a)*e^((d*x + c)*b/d - b*c/d)/b^2)/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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