3.3.33 \(\int (f x)^m (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^2 \, dx\) [233]

Optimal. Leaf size=1154 \[ -\frac {10 b^2 c^2 d^2 (f x)^{3+m} \sqrt {d-c^2 d x^2}}{f^3 (4+m)^3 (6+m)}-\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) (f x)^{3+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^3 (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {2 b^2 c^4 d^2 (f x)^{5+m} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{f^5 (6+m)^3 (1-c x) (1+c x)}-\frac {2 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (2+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {30 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (2+m)^2 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d^2 (f x)^{2+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (2+m) (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {10 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^4 (4+m)^2 (6+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 (f x)^{4+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^4 (4+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 (f x)^{6+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^6 (6+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 d^2 (f x)^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (6+m) \left (8+6 m+m^2\right )}+\frac {5 d (f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (4+m) (6+m)}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (6+m)}-\frac {30 b^2 c^2 d^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (4+m) (6+m) (1-c x) (1+c x)}-\frac {10 b^2 c^2 d^2 (10+3 m) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^3 (6+m) (1-c x) (1+c x)}-\frac {2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};c^2 x^2\right )}{f^3 (2+m) (3+m) (4+m)^2 (6+m)^3 (1-c x) (1+c x)}+\frac {15 d^3 \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 )}{(6+m) \left (8+6 m+m^2\right )} \]

[Out]

5*d*(f*x)^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/f/(4+m)/(6+m)+(f*x)^(1+m)*(-c^2*d*x^2+d)^(5/2)*(a+b*
arccosh(c*x))^2/f/(6+m)-10*b^2*c^2*d^2*(f*x)^(3+m)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^3/(6+m)-2*b^2*c^2*d^2*(m^2+1
5*m+52)*(f*x)^(3+m)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/f^3/(4+m)^2/(6+m)^3/(-c*x+1)/(c*x+1)+2*b^2*c^4*d^2*(f*x)
^(5+m)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/f^5/(6+m)^3/(-c*x+1)/(c*x+1)+15*d^2*(f*x)^(1+m)*(a+b*arccosh(c*x))^2*
(-c^2*d*x^2+d)^(1/2)/f/(6+m)/(m^2+6*m+8)-2*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+
m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-30*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(2+m)^
2/(4+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-10*b*c*d^2*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^2/(
6+m)/(m^2+6*m+8)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+10*b*c^3*d^2*(f*x)^(4+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/
f^4/(4+m)^2/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4*b*c^3*d^2*(f*x)^(4+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/
f^4/(4+m)/(6+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*b*c^5*d^2*(f*x)^(6+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/f^
6/(6+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-30*b^2*c^2*d^2*(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)/(6+m)/(-c*x+1)/(c*x+1)-10*b^2*c^2*d^2*(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/(6+m)/(m^2+5*m+6)/(-c*x+1)/(c*x+1)-2*b^2*c^2*d^2*(15*m^2+130*m+264)*(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)^2/(6+m)^3/(m^2+5*m+6)/(-c*x+1)/(c*x+
1)+15*d^3*Unintegrable((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)/(6+m)/(m^2+6*m+8)

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Rubi [A]
time = 1.47, 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 = {} \begin {gather*} \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

(-10*b^2*c^2*d^2*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3*(6 + m)) - (2*b^2*c^2*d^2*(52 + 15*m + m^2)
*(f*x)^(3 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^2*(6 + m)^3*(1 - c*x)*(1 + c*x)) + (2*b^2*c^4*d
^2*(f*x)^(5 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^5*(6 + m)^3*(1 - c*x)*(1 + c*x)) - (2*b*c*d^2*(f*x)^(2
+ m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (30*b*c*d^
2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) - (10*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(4 + m)*(6 + m)*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]) + (10*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 +
m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x
]))/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*(f*x)^(6 + m)*Sqrt[d - c^2*d*x^2]*(a + b
*ArcCosh[c*x]))/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x])^2)/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d*(f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])
^2)/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(f*(6 + m)) - (30*b^2*c
^2*d^2*(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)^2*(3 + m)*(4 + m)*(6 + m)*(1 - c*x)*(1 + c*x)) - (10*b^2*c^2*d^2*(10 + 3*m)*(f*x)^(3 + m)*Sqr
t[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*(6 + m)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*(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)^2*(6
+ m)^3*(1 - c*x)*(1 + c*x)) + (15*d^3*Defer[Int][((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/((
6 + m)*(8 + 6*m + m^2))

Rubi steps

\begin {align*} \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f x)^m (-1+c x)^{5/2} (1+c x)^{5/2} \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 = 0.96, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)

[Out]

int((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)^2*(f*x)^m, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcc
osh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)*(f*x)^m, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}\,{\left (f\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2)*(f*x)^m,x)

[Out]

int((a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2)*(f*x)^m, x)

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