Integrand size = 18, antiderivative size = 109 \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {2 (d x)^{7/2} (a+b \arccos (c x))^2}{7 d}+\frac {8 b c (d x)^{9/2} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {13}{4},c^2 x^2\right )}{63 d^2}+\frac {16 b^2 c^2 (d x)^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};c^2 x^2\right )}{693 d^3} \] Output:
2/7*(d*x)^(7/2)*(a+b*arccos(c*x))^2/d+8/63*b*c*(d*x)^(9/2)*(a+b*arccos(c*x ))*hypergeom([1/2, 9/4],[13/4],c^2*x^2)/d^2+16/693*b^2*c^2*(d*x)^(11/2)*hy pergeom([1, 11/4, 11/4],[13/4, 15/4],c^2*x^2)/d^3
Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(109)=218\).
Time = 10.85 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {(d x)^{5/2} \left (882 a^2 x^3+\frac {84 a b \left (-2 \sqrt {1-c^2 x^2} \left (5+3 c^2 x^2\right )+21 c^3 x^3 \arccos (c x)+10 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )\right )}{c^3}+\frac {b^2 \left (-16 c x \left (35+9 c^2 x^2\right )-168 \sqrt {1-c^2 x^2} \left (5+3 c^2 x^2\right ) \arccos (c x)+882 c^3 x^3 \arccos (c x)^2+840 \sqrt {1-c^2 x^2} \arccos (c x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {5}{4},c^2 x^2\right )+\frac {105 \sqrt {2} c \pi x \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{\operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {Gamma}\left (\frac {7}{4}\right )}\right )}{c^3}\right )}{3087 x^2} \] Input:
Integrate[(d*x)^(5/2)*(a + b*ArcCos[c*x])^2,x]
Output:
((d*x)^(5/2)*(882*a^2*x^3 + (84*a*b*(-2*Sqrt[1 - c^2*x^2]*(5 + 3*c^2*x^2) + 21*c^3*x^3*ArcCos[c*x] + 10*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2]))/ c^3 + (b^2*(-16*c*x*(35 + 9*c^2*x^2) - 168*Sqrt[1 - c^2*x^2]*(5 + 3*c^2*x^ 2)*ArcCos[c*x] + 882*c^3*x^3*ArcCos[c*x]^2 + 840*Sqrt[1 - c^2*x^2]*ArcCos[ c*x]*Hypergeometric2F1[3/4, 1, 5/4, c^2*x^2] + (105*Sqrt[2]*c*Pi*x*Hyperge ometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gamma[5/4]*Gamma[7/4])))/ c^3))/(3087*x^2)
Time = 0.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5139, 5221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle \frac {4 b c \int \frac {(d x)^{7/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{7 d}+\frac {2 (d x)^{7/2} (a+b \arccos (c x))^2}{7 d}\) |
\(\Big \downarrow \) 5221 |
\(\displaystyle \frac {4 b c \left (\frac {4 b c (d x)^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};c^2 x^2\right )}{99 d^2}+\frac {2 (d x)^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {13}{4},c^2 x^2\right ) (a+b \arccos (c x))}{9 d}\right )}{7 d}+\frac {2 (d x)^{7/2} (a+b \arccos (c x))^2}{7 d}\) |
Input:
Int[(d*x)^(5/2)*(a + b*ArcCos[c*x])^2,x]
Output:
(2*(d*x)^(7/2)*(a + b*ArcCos[c*x])^2)/(7*d) + (4*b*c*((2*(d*x)^(9/2)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, 9/4, 13/4, c^2*x^2])/(9*d) + (4*b*c* (d*x)^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, c^2*x^2])/(9 9*d^2)))/(7*d)
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[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]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. )*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*(a + b*ArcCos[c*x] )*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*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)))*S imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m /2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[m]
\[\int \left (d x \right )^{\frac {5}{2}} \left (a +b \arccos \left (c x \right )\right )^{2}d x\]
Input:
int((d*x)^(5/2)*(a+b*arccos(c*x))^2,x)
Output:
int((d*x)^(5/2)*(a+b*arccos(c*x))^2,x)
\[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*x)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
integral((b^2*d^2*x^2*arccos(c*x)^2 + 2*a*b*d^2*x^2*arccos(c*x) + a^2*d^2* x^2)*sqrt(d*x), x)
Timed out. \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Timed out} \] Input:
integrate((d*x)**(5/2)*(a+b*acos(c*x))**2,x)
Output:
Timed out
\[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*x)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
2/7*b^2*d^(5/2)*x^(7/2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 1/4 2*a^2*c^2*d^(5/2)*(4*(3*c^2*x^(7/2) + 7*x^(3/2))/c^4 + 42*arctan(sqrt(c)*s qrt(x))/c^(11/2) + 21*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^( 11/2)) + 14*a*b*c^2*d^(5/2)*integrate(1/7*x^(9/2)*arctan(sqrt(c*x + 1)*sqr t(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x) - 4*b^2*c*d^(5/2)*integrate(1/7*sqrt( c*x + 1)*sqrt(-c*x + 1)*x^(7/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x)) /(c^2*x^2 - 1), x) - 1/6*a^2*d^(5/2)*(4*x^(3/2)/c^2 + 6*arctan(sqrt(c)*sqr t(x))/c^(7/2) + 3*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(7/2) ) - 14*a*b*d^(5/2)*integrate(1/7*x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x)
Exception generated. \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((d*x)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d\,x\right )}^{5/2} \,d x \] Input:
int((a + b*acos(c*x))^2*(d*x)^(5/2),x)
Output:
int((a + b*acos(c*x))^2*(d*x)^(5/2), x)
\[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (2 \sqrt {x}\, a^{2} x^{3}+14 \left (\int \sqrt {x}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b +7 \left (\int \sqrt {x}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2}\right )}{7} \] Input:
int((d*x)^(5/2)*(a+b*acos(c*x))^2,x)
Output:
(sqrt(d)*d**2*(2*sqrt(x)*a**2*x**3 + 14*int(sqrt(x)*acos(c*x)*x**2,x)*a*b + 7*int(sqrt(x)*acos(c*x)**2*x**2,x)*b**2))/7