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=584 \[ \frac {3 d^2 \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}},x\right )}{m^2+6 m+8}-\frac {2 b c d \sqrt {d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2) (m+4) \sqrt {c x-1} \sqrt {c x+1}}-\frac {6 b c d \sqrt {d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \sqrt {d-c^2 d x^2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f \left (m^2+6 m+8\right )}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (m+4)}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2} (f x)^{m+4} \left (a+b \cosh ^{-1}(c x)\right )}{f^4 (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b^2 c^2 d (3 m+10) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right )}{f^3 (m+2) (m+3) (m+4)^3 (1-c x) (c x+1)}-\frac {6 b^2 c^2 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right )}{f^3 (m+2)^2 (m+3) (m+4) (1-c x) (c x+1)}-\frac {2 b^2 c^2 d \sqrt {d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^3} \]

[Out]

(f*x)^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/f/(4+m)-2*b^2*c^2*d*(f*x)^(3+m)*(-c^2*d*x^2+d)^(1/2)/f^3
/(4+m)^3+3*d*(f*x)^(1+m)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/f/(m^2+6*m+8)-6*b*c*d*(f*x)^(2+m)*(a+b*arcc
osh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)^2/(4+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*b*c*d*(f*x)^(2+m)*(a+b*arccosh(
c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)/(4+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2*b*c^3*d*(f*x)^(4+m)*(a+b*arccosh(c*x)
)*(-c^2*d*x^2+d)^(1/2)/f^4/(4+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-6*b^2*c^2*d*(f*x)^(3+m)*hypergeom([1/2, 3/2+1/2
*m],[5/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/f^3/(2+m)^2/(3+m)/(4+m)/(-c*x+1)/(c*x+1)-2*b^
2*c^2*d*(10+3*m)*(f*x)^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)
^(1/2)/f^3/(4+m)^3/(m^2+5*m+6)/(-c*x+1)/(c*x+1)+3*d^2*Unintegrable((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)
^(1/2),x)/(m^2+6*m+8)

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Rubi [A]  time = 0.52, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.53, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]

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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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \left (f x\right )^{m}, x\right ) \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 1.75, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}\,{\left (f\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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