\(\int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx\) [213]

Optimal result

Integrand size = 18, antiderivative size = 105 \[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=-\frac {2 (a+b \arccos (c x))^2}{d \sqrt {d x}}-\frac {8 b c \sqrt {d x} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )}{d^2}-\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{3 d^3} \]

[Out]

-16/3*b^2*c^2*(d*x)^(3/2)*hypergeom([3/4, 3/4, 1],[5/4, 7/4],c^2*x^2)/d^3-2*(a+b*arccos(c*x))^2/d/(d*x)^(1/2)-
8*b*c*(a+b*arccos(c*x))*hypergeom([1/4, 1/2],[5/4],c^2*x^2)*(d*x)^(1/2)/d^2

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4724, 4806} \[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=-\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{3 d^3}-\frac {8 b c \sqrt {d x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right ) (a+b \arccos (c x))}{d^2}-\frac {2 (a+b \arccos (c x))^2}{d \sqrt {d x}} \]

[In]

Int[(a + b*ArcCos[c*x])^2/(d*x)^(3/2),x]

[Out]

(-2*(a + b*ArcCos[c*x])^2)/(d*Sqrt[d*x]) - (8*b*c*Sqrt[d*x]*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/4, 1/2, 5/
4, c^2*x^2])/d^2 - (16*b^2*c^2*(d*x)^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(3*d^3)

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[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]

Rule 4806

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)))*Simp[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b \arccos (c x))^2}{d \sqrt {d x}}-\frac {(4 b c) \int \frac {a+b \arccos (c x)}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d} \\ & = -\frac {2 (a+b \arccos (c x))^2}{d \sqrt {d x}}-\frac {8 b c \sqrt {d x} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )}{d^2}-\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{3 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=\frac {x \left (-\frac {\sqrt {2} b^2 c^2 \pi x^2 \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{\operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {Gamma}\left (\frac {7}{4}\right )}-2 \left ((a+b \arccos (c x))^2+4 a b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )+2 b^2 \arccos (c x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {5}{4},c^2 x^2\right ) \sin (2 \arccos (c x))\right )\right )}{(d x)^{3/2}} \]

[In]

Integrate[(a + b*ArcCos[c*x])^2/(d*x)^(3/2),x]

[Out]

(x*(-((Sqrt[2]*b^2*c^2*Pi*x^2*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gamma[5/4]*Gamma[7/4]))
- 2*((a + b*ArcCos[c*x])^2 + 4*a*b*c*x*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2] + 2*b^2*ArcCos[c*x]*Hypergeom
etric2F1[3/4, 1, 5/4, c^2*x^2]*Sin[2*ArcCos[c*x]])))/(d*x)^(3/2)

Maple [F]

\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{2}}{\left (d x \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*arccos(c*x))^2/(d*x)^(3/2),x)

[Out]

int((a+b*arccos(c*x))^2/(d*x)^(3/2),x)

Fricas [F]

\[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arccos(c*x))^2/(d*x)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2)*sqrt(d*x)/(d^2*x^2), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*acos(c*x))**2/(d*x)**(3/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F]

\[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arccos(c*x))^2/(d*x)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(4*b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 - (a^2*c^2*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/(c^(3/2
)*d^2) + log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(c^(3/2)*d^2)) + 4*a*b*c^2*sqrt(d)*integrate(x^(5/2)
*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d^2*x^4 - d^2*x^2), x) + 8*b^2*c*sqrt(d)*integrate(sqrt(c*x +
1)*sqrt(-c*x + 1)*x^(3/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d^2*x^4 - d^2*x^2), x) - a^2*sqrt(d
)*(2*sqrt(c)*arctan(sqrt(c)*sqrt(x))/d^2 + sqrt(c)*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/d^2 + 4/(d
^2*sqrt(x))) - 4*a*b*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d^2*x^4 - d^2*x
^2), x))*d^(3/2)*sqrt(x))/(d^(3/2)*sqrt(x))

Giac [F]

\[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*arccos(c*x))^2/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccos(c*x) + a)^2/(d*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{(d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \]

[In]

int((a + b*acos(c*x))^2/(d*x)^(3/2),x)

[Out]

int((a + b*acos(c*x))^2/(d*x)^(3/2), x)