3.291 \(\int (c e+d e x)^{7/2} (a+b \sin ^{-1}(c+d x))^2 \, dx\)
Optimal. Leaf size=130 \[ \frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )}{1287 d e^3}-\frac {8 b (e (c+d x))^{11/2} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{99 d e^2}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d e} \]
[Out]
2/9*(e*(d*x+c))^(9/2)*(a+b*arcsin(d*x+c))^2/d/e-8/99*b*(e*(d*x+c))^(11/2)*(a+b*arcsin(d*x+c))*hypergeom([1/2,
11/4],[15/4],(d*x+c)^2)/d/e^2+16/1287*b^2*(e*(d*x+c))^(13/2)*HypergeometricPFQ([1, 13/4, 13/4],[15/4, 17/4],(d
*x+c)^2)/d/e^3
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Rubi [A] time = 0.21, antiderivative size = 130, 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 =
{4805, 4627, 4711} \[ \frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )}{1287 d e^3}-\frac {8 b (e (c+d x))^{11/2} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{99 d e^2}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d e} \]
Antiderivative was successfully verified.
[In]
Int[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x])^2,x]
[Out]
(2*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x])^2)/(9*d*e) - (8*b*(e*(c + d*x))^(11/2)*(a + b*ArcSin[c + d*x])*
Hypergeometric2F1[1/2, 11/4, 15/4, (c + d*x)^2])/(99*d*e^2) + (16*b^2*(e*(c + d*x))^(13/2)*HypergeometricPFQ[{
1, 13/4, 13/4}, {15/4, 17/4}, (c + d*x)^2])/(1287*d*e^3)
Rule 4627
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[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 4711
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)
^(m + 1)*(a + b*ArcSin[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[c^2*d + e, 0] && GtQ[d, 0] && !IntegerQ[m]
Rule 4805
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[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)^{7/2} \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{7/2} \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {(e x)^{9/2} \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d e}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{9 d e}-\frac {8 b (e (c+d x))^{11/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right )}{99 d e^2}+\frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )}{1287 d e^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 114, normalized size = 0.88 \[ \frac {2 e^3 (c+d x)^4 \sqrt {e (c+d x)} \left (8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};(c+d x)^2\right )+13 \left (a+b \sin ^{-1}(c+d x)\right ) \left (11 \left (a+b \sin ^{-1}(c+d x)\right )-4 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};(c+d x)^2\right )\right )\right )}{1287 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x])^2,x]
[Out]
(2*e^3*(c + d*x)^4*Sqrt[e*(c + d*x)]*(13*(a + b*ArcSin[c + d*x])*(11*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*H
ypergeometric2F1[1/2, 11/4, 15/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 13/4, 13/4}, {15/4,
17/4}, (c + d*x)^2]))/(1287*d)
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} d^{3} e^{3} x^{3} + 3 \, a^{2} c d^{2} e^{3} x^{2} + 3 \, a^{2} c^{2} d e^{3} x + a^{2} c^{3} e^{3} + {\left (b^{2} d^{3} e^{3} x^{3} + 3 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, b^{2} c^{2} d e^{3} x + b^{2} c^{3} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (a b d^{3} e^{3} x^{3} + 3 \, a b c d^{2} e^{3} x^{2} + 3 \, a b c^{2} d e^{3} x + a b c^{3} e^{3}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c))^2,x, algorithm="fricas")
[Out]
integral((a^2*d^3*e^3*x^3 + 3*a^2*c*d^2*e^3*x^2 + 3*a^2*c^2*d*e^3*x + a^2*c^3*e^3 + (b^2*d^3*e^3*x^3 + 3*b^2*c
*d^2*e^3*x^2 + 3*b^2*c^2*d*e^3*x + b^2*c^3*e^3)*arcsin(d*x + c)^2 + 2*(a*b*d^3*e^3*x^3 + 3*a*b*c*d^2*e^3*x^2 +
3*a*b*c^2*d*e^3*x + a*b*c^3*e^3)*arcsin(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 \[ \int {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c))^2,x, algorithm="giac")
[Out]
integrate((d*e*x + c*e)^(7/2)*(b*arcsin(d*x + c) + a)^2, x)
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {7}{2}} \left (a +b \arcsin \left (d x +c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c))^2,x)
[Out]
int((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c))^2,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c))^2,x, algorithm="maxima")
[Out]
1/630*(140*(b^2*d^4*e^3*x^4 + 4*b^2*c*d^3*e^3*x^3 + 6*b^2*c^2*d^2*e^3*x^2 + 4*b^2*c^3*d*e^3*x + b^2*c^4*e^3)*s
qrt(d*x + c)*sqrt(e)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + (11340*a*b*d^5*e^(7/2)*integra
te(1/9*sqrt(d*x + c)*x^5*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x -
c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 56700*a*b*c*d^4*e^(7/2)*integrate(1/9*sqrt(d*x + c)*x^4*arctan(d*
x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2
- 1), x) + 113400*a*b*c^2*d^3*e^(7/2)*integrate(1/9*sqrt(d*x + c)*x^3*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x
- c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 113400*a*b*c^3*d^2*e
^(7/2)*integrate(1/9*sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1
)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 56700*a*b*c^4*d*e^(7/2)*integrate(1/9*sqrt(d*x + c)
*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*
c*d*x + c^2 - 1), x) + 11340*a*b*c^5*e^(7/2)*integrate(1/9*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d
*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 315*a^2*c^5*e^(7/
2)*(2*arctan(sqrt(d*x + c)) - log(sqrt(d*x + c) + 1) + log(sqrt(d*x + c) - 1))/d + 2520*b^2*d^4*e^(7/2)*integr
ate(1/9*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*x^4*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1
)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 10080*b^2*c*d^3*e^(7/2)*int
egrate(1/9*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*x^3*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c
+ 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 15120*b^2*c^2*d^2*e^(7/2
)*integrate(1/9*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x
- c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 10080*b^2*c^3*d*e^(
7/2)*integrate(1/9*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*
x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 2520*b^2*c^4*e^(7/
2)*integrate(1/9*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x -
c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) - 1575*(2*(c + 1)*arctan
(sqrt(d*x + c)) - (c - 1)*log(sqrt(d*x + c) + 1) + (c - 1)*log(sqrt(d*x + c) - 1) - 4*sqrt(d*x + c))*a^2*c^4*e
^(7/2)/d - 11340*a*b*d^3*e^(7/2)*integrate(1/9*sqrt(d*x + c)*x^3*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c +
1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) - 34020*a*b*c*d^2*e^(7/2)*i
ntegrate(1/9*sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-
d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) - 34020*a*b*c^2*d*e^(7/2)*integrate(1/9*sqrt(d*x + c)*x*arcta
n(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x +
c^2 - 1), x) - 11340*a*b*c^3*e^(7/2)*integrate(1/9*sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c +
1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^2*x^2 + 2*c*d*x + c^2 - 1), x) + 1050*(6*(c^2 + 2*c + 1)*a
rctan(sqrt(d*x + c)) - 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) + 1) + 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) - 1) + 4
*(d*x + c)^(3/2) - 24*sqrt(d*x + c)*c)*a^2*c^3*e^(7/2)/d - 315*a^2*c^3*e^(7/2)*(2*arctan(sqrt(d*x + c)) - log(
sqrt(d*x + c) + 1) + log(sqrt(d*x + c) - 1))/d + 630*(4*(d*x + c)^(5/2) - 20*(d*x + c)^(3/2)*c - 10*(c^3 + 3*c
^2 + 3*c + 1)*arctan(sqrt(d*x + c)) + 5*(c^3 - 3*c^2 + 3*c - 1)*log(sqrt(d*x + c) + 1) - 5*(c^3 - 3*c^2 + 3*c
- 1)*log(sqrt(d*x + c) - 1) + 20*(3*c^2 + 1)*sqrt(d*x + c))*a^2*c^2*e^(7/2)/d + 945*(2*(c + 1)*arctan(sqrt(d*x
+ c)) - (c - 1)*log(sqrt(d*x + c) + 1) + (c - 1)*log(sqrt(d*x + c) - 1) - 4*sqrt(d*x + c))*a^2*c^2*e^(7/2)/d
+ 15*(60*(d*x + c)^(7/2) - 336*(d*x + c)^(5/2)*c + 140*(6*c^2 + 1)*(d*x + c)^(3/2) + 210*(c^4 + 4*c^3 + 6*c^2
+ 4*c + 1)*arctan(sqrt(d*x + c)) - 105*(c^4 - 4*c^3 + 6*c^2 - 4*c + 1)*log(sqrt(d*x + c) + 1) + 105*(c^4 - 4*c
^3 + 6*c^2 - 4*c + 1)*log(sqrt(d*x + c) - 1) - 1680*(c^3 + c)*sqrt(d*x + c))*a^2*c*e^(7/2)/d - 315*(6*(c^2 + 2
*c + 1)*arctan(sqrt(d*x + c)) - 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c) + 1) + 3*(c^2 - 2*c + 1)*log(sqrt(d*x + c)
- 1) + 4*(d*x + c)^(3/2) - 24*sqrt(d*x + c)*c)*a^2*c*e^(7/2)/d + (140*(d*x + c)^(9/2) - 900*(d*x + c)^(7/2)*c
+ 252*(10*c^2 + 1)*(d*x + c)^(5/2) - 2100*(2*c^3 + c)*(d*x + c)^(3/2) - 630*(c^5 + 5*c^4 + 10*c^3 + 10*c^2 +
5*c + 1)*arctan(sqrt(d*x + c)) + 315*(c^5 - 5*c^4 + 10*c^3 - 10*c^2 + 5*c - 1)*log(sqrt(d*x + c) + 1) - 315*(c
^5 - 5*c^4 + 10*c^3 - 10*c^2 + 5*c - 1)*log(sqrt(d*x + c) - 1) + 1260*(5*c^4 + 10*c^2 + 1)*sqrt(d*x + c))*a^2*
e^(7/2)/d - 63*(4*(d*x + c)^(5/2) - 20*(d*x + c)^(3/2)*c - 10*(c^3 + 3*c^2 + 3*c + 1)*arctan(sqrt(d*x + c)) +
5*(c^3 - 3*c^2 + 3*c - 1)*log(sqrt(d*x + c) + 1) - 5*(c^3 - 3*c^2 + 3*c - 1)*log(sqrt(d*x + c) - 1) + 20*(3*c^
2 + 1)*sqrt(d*x + c))*a^2*e^(7/2)/d)*d)/d
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*e + d*e*x)^(7/2)*(a + b*asin(c + d*x))^2,x)
[Out]
int((c*e + d*e*x)^(7/2)*(a + b*asin(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)**(7/2)*(a+b*asin(d*x+c))**2,x)
[Out]
Timed out
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