3.234 \(\int (f x)^m (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=583 \[ \frac{3 d^2 \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}},x\right )}{m^2+6 m+8}-\frac{2 b^2 c^2 d (3 m+10) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right )}{f^3 (m+2) (m+3) (m+4)^3 (1-c x) (c x+1)}-\frac{6 b^2 c^2 d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right )}{f^3 (m+2)^2 (m+3) (m+4) (1-c x) (c x+1)}-\frac{2 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2) (m+4) \sqrt{c x-1} \sqrt{c x+1}}-\frac{6 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2} (f x)^{m+4} \left (a+b \cosh ^{-1}(c x)\right )}{f^4 (m+4)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 d \sqrt{d-c^2 d x^2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f \left (m^2+6 m+8\right )}+\frac{\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (m+4)}-\frac{2 b^2 c^2 d \sqrt{d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^3} \]

[Out]

(-2*b^2*c^2*d*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3) - (6*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*
(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*(f*x)^(2 + m)*Sqrt[d - c
^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*(f*x)^(4 + m)*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*(f*x)^(1 + m)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*Ar
cCosh[c*x])^2)/(f*(4 + m)) - (6*b^2*c^2*d*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F
1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)^2*(3 + m)*(4 + m)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d*(10
+ 3*m)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^
2])/(f^3*(2 + m)*(3 + m)*(4 + m)^3*(1 - c*x)*(1 + c*x)) + (3*d^2*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)
/Sqrt[d - c^2*d*x^2], x])/(8 + 6*m + m^2)

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Rubi [A]  time = 0.523117, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.513126, size = 0, normalized size = 0. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x]

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Maple [A]  time = 1.082, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x)

[Out]

int((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)^2*(f*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [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*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**2,x)

[Out]

Timed out

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

[Out]

Timed out