3.237 \(\int (c e+d e x)^{5/2} (a+b \sinh ^{-1}(c+d x))^2 \, dx\)

Optimal. Leaf size=134 \[ \frac {16 b^2 (e (c+d x))^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};-(c+d x)^2\right )}{693 d e^3}-\frac {8 b (e (c+d x))^{9/2} \, _2F_1\left (\frac {1}{2},\frac {9}{4};\frac {13}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{63 d e^2}+\frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{7 d e} \]

[Out]

2/7*(e*(d*x+c))^(7/2)*(a+b*arcsinh(d*x+c))^2/d/e-8/63*b*(e*(d*x+c))^(9/2)*(a+b*arcsinh(d*x+c))*hypergeom([1/2,
9/4],[13/4],-(d*x+c)^2)/d/e^2+16/693*b^2*(e*(d*x+c))^(11/2)*HypergeometricPFQ([1, 11/4, 11/4],[13/4, 15/4],-(
d*x+c)^2)/d/e^3

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Rubi [A]  time = 0.22, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5865, 5661, 5762} \[ \frac {16 b^2 (e (c+d x))^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};-(c+d x)^2\right )}{693 d e^3}-\frac {8 b (e (c+d x))^{9/2} \, _2F_1\left (\frac {1}{2},\frac {9}{4};\frac {13}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{63 d e^2}+\frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{7 d e} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^2,x]

[Out]

(2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^2)/(7*d*e) - (8*b*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])
*Hypergeometric2F1[1/2, 9/4, 13/4, -(c + d*x)^2])/(63*d*e^2) + (16*b^2*(e*(c + d*x))^(11/2)*HypergeometricPFQ[
{1, 11/4, 11/4}, {13/4, 15/4}, -(c + d*x)^2])/(693*d*e^3)

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5762

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x
)^(m + 1)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/(Sqrt[d]*f*(m + 1)),
x] - Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/(Sqrt
[d]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] &&  !IntegerQ[m]

Rule 5865

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

\begin {align*} \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{5/2} \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{7 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {(e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{7 d e}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{7 d e}-\frac {8 b (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {9}{4};\frac {13}{4};-(c+d x)^2\right )}{63 d e^2}+\frac {16 b^2 (e (c+d x))^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};-(c+d x)^2\right )}{693 d e^3}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 110, normalized size = 0.82 \[ \frac {2 (e (c+d x))^{7/2} \left (8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};-(c+d x)^2\right )-44 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {9}{4};\frac {13}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )+99 \left (a+b \sinh ^{-1}(c+d x)\right )^2\right )}{693 d e} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^2,x]

[Out]

(2*(e*(c + d*x))^(7/2)*(99*(a + b*ArcSinh[c + d*x])^2 - 44*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric
2F1[1/2, 9/4, 13/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, -(c + d
*x)^2]))/(693*d*e)

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} d^{2} e^{2} x^{2} + 2 \, a^{2} c d e^{2} x + a^{2} c^{2} e^{2} + {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + b^{2} c^{2} e^{2}\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 2 \, {\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x + a b c^{2} e^{2}\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral((a^2*d^2*e^2*x^2 + 2*a^2*c*d*e^2*x + a^2*c^2*e^2 + (b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x + b^2*c^2*e^2)*
arcsinh(d*x + c)^2 + 2*(a*b*d^2*e^2*x^2 + 2*a*b*c*d*e^2*x + a*b*c^2*e^2)*arcsinh(d*x + c))*sqrt(d*e*x + c*e),
x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(5/2)*(b*arcsinh(d*x + c) + a)^2, x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^2,x)

[Out]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (d e x + c e\right )}^{\frac {7}{2}} a^{2}}{7 \, d e} + \frac {2 \, {\left (b^{2} d^{3} e^{\frac {5}{2}} x^{3} + 3 \, b^{2} c d^{2} e^{\frac {5}{2}} x^{2} + 3 \, b^{2} c^{2} d e^{\frac {5}{2}} x + b^{2} c^{3} e^{\frac {5}{2}}\right )} \sqrt {d x + c} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2}}{7 \, d} + \int -\frac {2 \, {\left ({\left (2 \, b^{2} c^{4} e^{\frac {5}{2}} - {\left (7 \, a b d^{4} e^{\frac {5}{2}} - 2 \, b^{2} d^{4} e^{\frac {5}{2}}\right )} x^{4} - 4 \, {\left (7 \, a b c d^{3} e^{\frac {5}{2}} - 2 \, b^{2} c d^{3} e^{\frac {5}{2}}\right )} x^{3} - 7 \, {\left (c^{4} e^{\frac {5}{2}} + c^{2} e^{\frac {5}{2}}\right )} a b + {\left (12 \, b^{2} c^{2} d^{2} e^{\frac {5}{2}} - 7 \, {\left (6 \, c^{2} d^{2} e^{\frac {5}{2}} + d^{2} e^{\frac {5}{2}}\right )} a b\right )} x^{2} + 2 \, {\left (4 \, b^{2} c^{3} d e^{\frac {5}{2}} - 7 \, {\left (2 \, c^{3} d e^{\frac {5}{2}} + c d e^{\frac {5}{2}}\right )} a b\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} - {\left ({\left (7 \, a b d^{5} e^{\frac {5}{2}} - 2 \, b^{2} d^{5} e^{\frac {5}{2}}\right )} x^{5} + 5 \, {\left (7 \, a b c d^{4} e^{\frac {5}{2}} - 2 \, b^{2} c d^{4} e^{\frac {5}{2}}\right )} x^{4} + {\left (7 \, {\left (10 \, c^{2} d^{3} e^{\frac {5}{2}} + d^{3} e^{\frac {5}{2}}\right )} a b - 2 \, {\left (10 \, c^{2} d^{3} e^{\frac {5}{2}} + d^{3} e^{\frac {5}{2}}\right )} b^{2}\right )} x^{3} + 7 \, {\left (c^{5} e^{\frac {5}{2}} + c^{3} e^{\frac {5}{2}}\right )} a b - 2 \, {\left (c^{5} e^{\frac {5}{2}} + c^{3} e^{\frac {5}{2}}\right )} b^{2} + {\left (7 \, {\left (10 \, c^{3} d^{2} e^{\frac {5}{2}} + 3 \, c d^{2} e^{\frac {5}{2}}\right )} a b - 2 \, {\left (10 \, c^{3} d^{2} e^{\frac {5}{2}} + 3 \, c d^{2} e^{\frac {5}{2}}\right )} b^{2}\right )} x^{2} + {\left (7 \, {\left (5 \, c^{4} d e^{\frac {5}{2}} + 3 \, c^{2} d e^{\frac {5}{2}}\right )} a b - 2 \, {\left (5 \, c^{4} d e^{\frac {5}{2}} + 3 \, c^{2} d e^{\frac {5}{2}}\right )} b^{2}\right )} x\right )} \sqrt {d x + c}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}{7 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (3 \, c^{2} d + d\right )} x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")

[Out]

2/7*(d*e*x + c*e)^(7/2)*a^2/(d*e) + 2/7*(b^2*d^3*e^(5/2)*x^3 + 3*b^2*c*d^2*e^(5/2)*x^2 + 3*b^2*c^2*d*e^(5/2)*x
+ b^2*c^3*e^(5/2))*sqrt(d*x + c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2/d + integrate(-2/7*((2*b^
2*c^4*e^(5/2) - (7*a*b*d^4*e^(5/2) - 2*b^2*d^4*e^(5/2))*x^4 - 4*(7*a*b*c*d^3*e^(5/2) - 2*b^2*c*d^3*e^(5/2))*x^
3 - 7*(c^4*e^(5/2) + c^2*e^(5/2))*a*b + (12*b^2*c^2*d^2*e^(5/2) - 7*(6*c^2*d^2*e^(5/2) + d^2*e^(5/2))*a*b)*x^2
+ 2*(4*b^2*c^3*d*e^(5/2) - 7*(2*c^3*d*e^(5/2) + c*d*e^(5/2))*a*b)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d
*x + c) - ((7*a*b*d^5*e^(5/2) - 2*b^2*d^5*e^(5/2))*x^5 + 5*(7*a*b*c*d^4*e^(5/2) - 2*b^2*c*d^4*e^(5/2))*x^4 + (
7*(10*c^2*d^3*e^(5/2) + d^3*e^(5/2))*a*b - 2*(10*c^2*d^3*e^(5/2) + d^3*e^(5/2))*b^2)*x^3 + 7*(c^5*e^(5/2) + c^
3*e^(5/2))*a*b - 2*(c^5*e^(5/2) + c^3*e^(5/2))*b^2 + (7*(10*c^3*d^2*e^(5/2) + 3*c*d^2*e^(5/2))*a*b - 2*(10*c^3
*d^2*e^(5/2) + 3*c*d^2*e^(5/2))*b^2)*x^2 + (7*(5*c^4*d*e^(5/2) + 3*c^2*d*e^(5/2))*a*b - 2*(5*c^4*d*e^(5/2) + 3
*c^2*d*e^(5/2))*b^2)*x)*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^3*x^3 + 3*c*d^2*x^2
+ c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(5/2)*(a + b*asinh(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)^(5/2)*(a + b*asinh(c + d*x))^2, 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((d*e*x+c*e)**(5/2)*(a+b*asinh(d*x+c))**2,x)

[Out]

Timed out

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