3.160 \(\int \frac{(f x)^m (a+b \cosh ^{-1}(c x))}{\sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}} \, dx\)

Optimal. Leaf size=188 \[ \frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},c^2 x^2\right )}{f^2 (m+1) (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{\sqrt{1-c^2 x^2} (f x)^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}} \]

[Out]

((f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(
f*(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometri
cPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c
*d2*x])

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Rubi [A]  time = 0.558932, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {5765, 5763} \[ \frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{\sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}} \]

Antiderivative was successfully verified.

[In]

Int[((f*x)^m*(a + b*ArcCosh[c*x]))/(Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]),x]

[Out]

((f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(
f*(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometri
cPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c
*d2*x])

Rule 5765

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + c*x]*Sqrt[-1 + c*x])/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[((f*x)^m*(a
+ b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && E
qQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !(GtQ[d1, 0] && LtQ[d2, 0]) && (IntegerQ[m] || EqQ[n, 1
])

Rule 5763

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_
)]), x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, c^2*x^2])/(f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Simp[(b*c*(f*x)^(m + 2)*Hypergeometric
PFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[-(d1*d2)]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{
a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] &&  !
IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}\\ &=\frac{(f x)^{1+m} \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};c^2 x^2\right )}{f (1+m) \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}+\frac{b c (f x)^{2+m} \sqrt{-1+c x} \sqrt{1+c x} \, _3F_2\left (1,1+\frac{m}{2},1+\frac{m}{2};\frac{3}{2}+\frac{m}{2},2+\frac{m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}}\\ \end{align*}

Mathematica [C]  time = 6.0086, size = 264, normalized size = 1.4 \[ \frac{2^{-m-3} \sqrt{c \text{d1} x+\text{d1}} \left (\frac{c x}{c x+1}\right )^{1-m} (f x)^m \left (b m \left (\frac{c x}{c x+1}\right )^m \sinh \left (2 \cosh ^{-1}(c x)\right ) \left (\sqrt{\pi } c (m+1) x \sqrt{\frac{c x-1}{c x+1}} \text{Gamma}(m+1) \, _3\tilde{F}_2\left (1,\frac{m+2}{2},\frac{m+2}{2};\frac{m+3}{2},\frac{m+4}{2};c^2 x^2\right )-2^{m+2} (c x-1) \cosh ^{-1}(c x) \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+3}{2},c^2 x^2\right )\right )+a 2^{m+3} (m+1) (c x-1) F_1\left (-m;-m,\frac{1}{2};1-m;\frac{1}{c x+1},\frac{2}{c x+1}\right )\right )}{c^2 \text{d1} m (m+1) x \sqrt{\frac{c x-1}{c x+1}} \sqrt{\text{d2}-c \text{d2} x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]),x]

[Out]

(2^(-3 - m)*(f*x)^m*((c*x)/(1 + c*x))^(1 - m)*Sqrt[d1 + c*d1*x]*(2^(3 + m)*a*(1 + m)*(-1 + c*x)*AppellF1[-m, -
m, 1/2, 1 - m, (1 + c*x)^(-1), 2/(1 + c*x)] + b*m*((c*x)/(1 + c*x))^m*(-(2^(2 + m)*(-1 + c*x)*ArcCosh[c*x]*Hyp
ergeometric2F1[1, (2 + m)/2, (3 + m)/2, c^2*x^2]) + c*(1 + m)*Sqrt[Pi]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Gamma[1 +
m]*HypergeometricPFQRegularized[{1, (2 + m)/2, (2 + m)/2}, {(3 + m)/2, (4 + m)/2}, c^2*x^2])*Sinh[2*ArcCosh[c*
x]]))/(c^2*d1*m*(1 + m)*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d2 - c*d2*x])

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Maple [F]  time = 0.505, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ){\frac{1}{\sqrt{c{\it d1}\,x+{\it d1}}}}{\frac{1}{\sqrt{-c{\it d2}\,x+{\it d2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x)

[Out]

int((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt{c d_{1} x + d_{1}} \sqrt{-c d_{2} x + d_{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)*(f*x)^m/(sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{c d_{1} x + d_{1}} \sqrt{-c d_{2} x + d_{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{c^{2} d_{1} d_{2} x^{2} - d_{1} d_{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(b*arccosh(c*x) + a)*(f*x)^m/(c^2*d1*d2*x^2 - d1*d2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f x\right )^{m} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\sqrt{d_{1} \left (c x + 1\right )} \sqrt{- d_{2} \left (c x - 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(a+b*acosh(c*x))/(c*d1*x+d1)**(1/2)/(-c*d2*x+d2)**(1/2),x)

[Out]

Integral((f*x)**m*(a + b*acosh(c*x))/(sqrt(d1*(c*x + 1))*sqrt(-d2*(c*x - 1))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt{c d_{1} x + d_{1}} \sqrt{-c d_{2} x + d_{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*(f*x)^m/(sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)), x)