3.521 \(\int (f x)^m (d+e x^2) (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=198 \[ -\frac{b \sqrt{1-c^2 x^2} (f x)^{m+2} \left (c^2 d (m+3)^2+e (m+1) (m+2)\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{c f^2 (m+1) (m+2) (m+3)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}+\frac{e (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}-\frac{b e \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2}}{c f^2 (m+3)^2} \]

[Out]

-((b*e*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*f^2*(3 + m)^2)) + (d*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))
/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) - (b*(e*(1 + m)*(2 + m) + c^2*d*(3 + m)^2)
*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c*f^2*(1 + m)*(2 + m)
*(3 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.20317, antiderivative size = 187, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5786, 460, 126, 365, 364} \[ \frac{d (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}+\frac{e (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}-\frac{b \sqrt{1-c^2 x^2} (f x)^{m+2} \left (\frac{c^2 d}{m^2+3 m+2}+\frac{e}{(m+3)^2}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{c f^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2}}{c f^2 (m+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*e*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*f^2*(3 + m)^2)) + (d*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))
/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) - (b*(e/(3 + m)^2 + (c^2*d)/(2 + 3*m + m^2
))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c*f^2*Sqrt[-1 + c*x
]*Sqrt[1 + c*x])

Rule 5786

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(d*(f*x)^(
m + 1)*(a + b*ArcCosh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c)/(f*(m + 1)*(m + 3)), Int[((f*x)^(m + 1)*(d*(m + 3)
+ e*(m + 1)*x^2))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] + Simp[(e*(f*x)^(m + 3)*(a + b*ArcCosh[c*x]))/(f^3*(
m + 3)), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[((a + b*x)^Fra
cPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a
, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac{(b c) \int \frac{(f x)^{1+m} \left (d (3+m)+e (1+m) x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{f \left (3+4 m+m^2\right )}\\ &=-\frac{b e (f x)^{2+m} \sqrt{-1+c x} \sqrt{1+c x}}{c f^2 (3+m)^2}+\frac{d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac{\left (b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right )\right ) \int \frac{(f x)^{1+m}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c f (1+m) (3+m)^2}\\ &=-\frac{b e (f x)^{2+m} \sqrt{-1+c x} \sqrt{1+c x}}{c f^2 (3+m)^2}+\frac{d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac{\left (b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\sqrt{-1+c^2 x^2}} \, dx}{c f (1+m) (3+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b e (f x)^{2+m} \sqrt{-1+c x} \sqrt{1+c x}}{c f^2 (3+m)^2}+\frac{d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac{\left (b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\sqrt{1-c^2 x^2}} \, dx}{c f (1+m) (3+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b e (f x)^{2+m} \sqrt{-1+c x} \sqrt{1+c x}}{c f^2 (3+m)^2}+\frac{d (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}-\frac{b \left (c^2 d (3+m)^2+e \left (2+3 m+m^2\right )\right ) (f x)^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{c f^2 (1+m) (2+m) (3+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.618508, size = 186, normalized size = 0.94 \[ x (f x)^m \left (\frac{\frac{\left (d (m+3)+e (m+1) x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{m+1}-\frac{b c e x^3 \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+4}{2},\frac{m+6}{2},c^2 x^2\right )}{(m+4) \sqrt{c x-1} \sqrt{c x+1}}}{m+3}-\frac{b c d x \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt{c x-1} \sqrt{c x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(f*x)^m*(-((b*c*d*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2
)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (((d*(3 + m) + e*(1 + m)*x^2)*(a + b*ArcCosh[c*x]))/(1 + m) - (b*c*e*x^3*Sq
rt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
/(3 + m))

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Maple [F]  time = 3.49, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(e*x^2+d)*(a+b*arccosh(c*x)),x)

[Out]

int((f*x)^m*(e*x^2+d)*(a+b*arccosh(c*x)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccosh(c*x))*(f*x)^m, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{m} \left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(e*x**2+d)*(a+b*acosh(c*x)),x)

[Out]

Integral((f*x)**m*(a + b*acosh(c*x))*(d + e*x**2), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out