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

Optimal. Leaf size=298 \[ -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c d e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]

[Out]

-2/3*(a+b*arccsc(c*x))/e/(e*x+d)^(3/2)+4/3*b*e*(-c^2*x^2+1)/c/d/(c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1
/2)-4/3*b*EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(
c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)+4/3*b*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(
1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5227, 1574, 958, 745, 21, 719, 424, 933, 168, 538, 537} \[ -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c d e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*e*(1 - c^2*x^2))/(3*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsc[c*x]))/
(3*e*(d + e*x)^(3/2)) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c
*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[(c*(d + e*x)
)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*d*e*Sqrt[1
- 1/(c^2*x^2)]*x*Sqrt[d + e*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 958

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegra
nd[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[n + 1/2]

Rule 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^
(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 5227

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + b
*ArcCsc[c*x]))/(e*(m + 1)), x] + Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x],
x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx &=-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {(2 b) \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e}\\ &=-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {\left (4 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {\left (4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}\\ &=\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [B]  time = 13.64, size = 725, normalized size = 2.43 \[ \frac {b \left (\frac {2 (c x)^{5/2} \left (\frac {d}{x}+e\right )^{5/2} \left (\frac {2 c d \sqrt {1-c^2 x^2} \sqrt {\frac {c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} (c x)^{3/2} \sqrt {\frac {d}{x}+e}}-\frac {2 e \cos \left (2 \csc ^{-1}(c x)\right ) \left (\left (c^2 x^2-1\right ) (c d+c e x)+c^2 d x \sqrt {1-c^2 x^2} \sqrt {\frac {c d+c e x}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )+c e x \sqrt {1-c^2 x^2} \sqrt {\frac {c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )-\frac {c x (c x+1) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e F\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (c x+1)}{e-c d}}}\right )}{c d \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c x} \left (c^2 x^2-2\right ) \sqrt {\frac {d}{x}+e}}\right )}{3 e (c d-e) (c d+e) (d+e x)^{5/2}}-\frac {c^3 x^3 \left (\frac {d}{x}+e\right )^3 \left (-\frac {4 \sqrt {1-\frac {1}{c^2 x^2}}}{3 c d \left (c^2 d^2-e^2\right )}-\frac {4 \left (c^2 d^2 \csc ^{-1}(c x)-c d e \sqrt {1-\frac {1}{c^2 x^2}}-e^2 \csc ^{-1}(c x)\right )}{3 c^2 d^2 \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )}+\frac {2 \csc ^{-1}(c x)}{3 c^2 d^2 e}+\frac {2 e \csc ^{-1}(c x)}{3 c^2 d^2 \left (\frac {d}{x}+e\right )^2}\right )}{(d+e x)^{5/2}}\right )}{c}-\frac {2 a}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a)/(3*e*(d + e*x)^(3/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 - 1/(c^2*x^2)])/(3*c*d*(c^2*d^2 - e^2))
+ (2*ArcCsc[c*x])/(3*c^2*d^2*e) + (2*e*ArcCsc[c*x])/(3*c^2*d^2*(e + d/x)^2) - (4*(-(c*d*e*Sqrt[1 - 1/(c^2*x^2)
]) + c^2*d^2*ArcCsc[c*x] - e^2*ArcCsc[c*x]))/(3*c^2*d^2*(c^2*d^2 - e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e
+ d/x)^(5/2)*(c*x)^(5/2)*((2*c*d*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 -
c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*e*Cos[2*ArcCsc[c*x]]*
((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[
1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]
*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c
*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c
*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*Sqrt[1 - 1/(c^
2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*(c*d - e)*e*(c*d + e)*(d + e*x)^(5/2))))/c

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 886, normalized size = 2.97 \[ \frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {c}{d c -e}}\, \left (e x +d \right )^{2} c^{2} d -\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \frac {d c -e}{c d}, \frac {\sqrt {\frac {c}{d c +e}}}{\sqrt {\frac {c}{d c -e}}}\right ) \sqrt {e x +d}\, c^{2} d^{2}-\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}+\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}-2 \sqrt {\frac {c}{d c -e}}\, \left (e x +d \right ) c^{2} d^{2}-\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c d e +\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c d e +\sqrt {\frac {c}{d c -e}}\, c^{2} d^{3}+\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \frac {d c -e}{c d}, \frac {\sqrt {\frac {c}{d c +e}}}{\sqrt {\frac {c}{d c -e}}}\right ) \sqrt {e x +d}\, e^{2}-\sqrt {\frac {c}{d c -e}}\, d \,e^{2}\right )}{3 c \left (d c -e \right ) \sqrt {e x +d}\, \left (d c +e \right ) \sqrt {\frac {c}{d c -e}}\, d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arccsc(c*x)-2/3/c*((c/(c*d-e))^(1/2)*(e*x+d)^2*c^2*d-(-((e*x+d
)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c
*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*(e*x+d)^(1/2)*c^2*d^2-(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+
d)*c-d*c-e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*(e*x+d)^(1/2)*c^
2*d^2+(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e)
)^(1/2),((c*d-e)/(c*d+e))^(1/2))*(e*x+d)^(1/2)*c^2*d^2-2*(c/(c*d-e))^(1/2)*(e*x+d)*c^2*d^2-(-((e*x+d)*c-d*c+e)
/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))
^(1/2))*(e*x+d)^(1/2)*c*d*e+(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*EllipticE((e
*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*(e*x+d)^(1/2)*c*d*e+(c/(c*d-e))^(1/2)*c^2*d^3+(-((e*x+d
)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c
*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*(e*x+d)^(1/2)*e^2-(c/(c*d-e))^(1/2)*d*e^2)/(c*d-e)/(e*x+d)^(1/2)/
(c*d+e)/(c/(c*d-e))^(1/2)/d^2/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-e>0)', see `assume?` for m
ore details)Is c*d-e positive or negative?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/(d + e*x)^(5/2),x)

[Out]

int((a + b*asin(1/(c*x)))/(d + e*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsc(c*x))/(d + e*x)**(5/2), x)

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