3.2.62 \(\int \frac {(f x)^m (a+b \cosh ^{-1}(c x))}{(\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2}} \, dx\) [162]

Optimal. Leaf size=504 \[ \frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {(2-m) m (f x)^{1+m} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f (1+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (2-m) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {b c (2-m) m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (1+m) (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \]

[Out]

1/3*(f*x)^(1+m)*(a+b*arccosh(c*x))/d1/d2/f/(c*d1*x+d1)^(3/2)/(-c*d2*x+d2)^(3/2)+1/3*(2-m)*(f*x)^(1+m)*(a+b*arc
cosh(c*x))/d1^2/d2^2/f/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2)+1/3*b*c*(2-m)*(f*x)^(2+m)*hypergeom([1, 1+1/2*m],[
2+1/2*m],c^2*x^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d1^2/d2^2/f^2/(2+m)/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2)+1/3*b*c
*(f*x)^(2+m)*hypergeom([2, 1+1/2*m],[2+1/2*m],c^2*x^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d1^2/d2^2/f^2/(2+m)/(c*d1*x
+d1)^(1/2)/(-c*d2*x+d2)^(1/2)-1/3*b*c*(2-m)*m*(f*x)^(2+m)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2*m, 3/2+1/2*m]
,c^2*x^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d1^2/d2^2/f^2/(1+m)/(2+m)/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2)-1/3*(2-m)
*m*(f*x)^(1+m)*(a+b*arccosh(c*x))*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/d1^2/d2^2
/f/(1+m)/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2)

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Rubi [A]
time = 0.84, antiderivative size = 504, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {5937, 5949, 74, 371} \begin {gather*} -\frac {b c (2-m) m \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 )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+1) (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}-\frac {(2-m) m \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 )}{3 \text {d1}^2 \text {d2}^2 f (m+1) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(2-m) (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {b c (2-m) \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _2F_1\left (2,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (m+2) \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)),x]

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(3*d1*d2*f*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)) + ((2 - m)*(f*x)^(1 +
m)*(a + b*ArcCosh[c*x]))/(3*d1^2*d2^2*f*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) - ((2 - m)*m*(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])/(3*d1^2*d2^2*f*(1 + m
)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(2 - m)*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometri
c2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d1^2*d2^2*f^2*(2 + 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]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d1^2*d2^2*f^
2*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) - (b*c*(2 - m)*m*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hyp
ergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(3*d1^2*d2^2*f^2*(1 + m)*(2 + m)*Sqrt[d1
+ c*d1*x]*Sqrt[d2 - c*d2*x])

Rule 74

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] ||  !IntegerQ[p])))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 5937

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

Rule 5949

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

Rubi steps

\begin {align*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2}} \, dx &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx}{3 \text {d1} \text {d2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{1+m}}{\left (-1+c^2 x^2\right )^2} \, dx}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \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} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {((2-m) m) \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}{3 \text {d1}^2 \text {d2}^2}-\frac {\left (b c (2-m) \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{1+m}}{-1+c^2 x^2} \, dx}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (2-m) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+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} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {\left ((2-m) m \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}{3 \text {d1}^2 \text {d2}^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1} \text {d2} f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {(2-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{3 \text {d1}^2 \text {d2}^2 f \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {(2-m) m (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 )}{3 \text {d1}^2 \text {d2}^2 f (1+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {b c (2-m) (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+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} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 \text {d1}^2 \text {d2}^2 f^2 (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {b c (2-m) m (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 )}{3 \text {d1}^2 \text {d2}^2 f^2 (1+m) (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}\\ \end {align*}

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Mathematica [F]
time = 1.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{(\text {d1}+c \text {d1} x)^{5/2} (\text {d2}-c \text {d2} x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)),x]

[Out]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)), x]

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{\left (c \mathit {d1} x +\mathit {d1} \right )^{\frac {5}{2}} \left (-c \mathit {d2} x +\mathit {d2} \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(5/2)/(-c*d2*x+d2)^(5/2),x)

[Out]

int((f*x)^m*(a+b*arccosh(c*x))/(c*d1*x+d1)^(5/2)/(-c*d2*x+d2)^(5/2),x)

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Maxima [F]
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*(a+b*arccosh(c*x))/(c*d1*x+d1)^(5/2)/(-c*d2*x+d2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)*(f*x)^m/((c*d1*x + d1)^(5/2)*(-c*d2*x + d2)^(5/2)), x)

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Fricas [F]
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*(a+b*arccosh(c*x))/(c*d1*x+d1)^(5/2)/(-c*d2*x+d2)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(b*arccosh(c*x) + a)*(f*x)^m/(c^6*d1^3*d2^3*x^6 - 3*c^4*d1^3*d2
^3*x^4 + 3*c^2*d1^3*d2^3*x^2 - d1^3*d2^3), 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*(a+b*acosh(c*x))/(c*d1*x+d1)**(5/2)/(-c*d2*x+d2)**(5/2),x)

[Out]

Timed out

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Giac [F]
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*(a+b*arccosh(c*x))/(c*d1*x+d1)^(5/2)/(-c*d2*x+d2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*(f*x)^m/((c*d1*x + d1)^(5/2)*(-c*d2*x + d2)^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(f*x)^m)/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)),x)

[Out]

int(((a + b*acosh(c*x))*(f*x)^m)/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)), x)

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