3.84 \(\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{e \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.177725, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6288, 932, 168, 538, 537} \[ \frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{e \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6288

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

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], 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 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 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 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)
])

Rubi steps

\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{e}\\ &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{e}\\ &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{\left (4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\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 )}{e}\\ &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{\left (4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{\frac{c (d+e x)}{c d+e}}\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 )}{e \sqrt{d+e x}}\\ &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \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 )}{e \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 10.3983, size = 1675, normalized size = 15.95 \[ -\frac{2 a}{e \sqrt{d+e x}}-\frac{2 b \text{sech}^{-1}(c x)}{e \sqrt{d+e x}}+\frac{4 i b \left (2 \sqrt{-\frac{i \left (c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \sqrt{-\frac{i \left (-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \left (\frac{1-c x}{c x+1}+1\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}}\right ),\frac{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right )^2}{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right )^2}\right )+\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \sqrt{\frac{1-c x}{c x+1}+1} \sqrt{\frac{-\frac{(1-c x) e}{c x+1}+e+c d \left (\frac{1-c x}{c x+1}+1\right )}{c d+e}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{1-c x}{c x+1}}\right ),\frac{c d-e}{c d+e}\right )+2 i \sqrt{-\frac{i \left (c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \sqrt{-\frac{i \left (-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \left (\frac{1-c x}{c x+1}+1\right ) \left (\Pi \left (\frac{i \sqrt{-c d-e}-\sqrt{c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}}\right )|\frac{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right )^2}{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right )^2}\right )-\Pi \left (\frac{\sqrt{c d-e}-i \sqrt{-c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}}\right )|\frac{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right )^2}{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right )^2}\right )\right )\right )}{e \sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \left (\frac{1-c x}{c x+1}+1\right ) \sqrt{\frac{c d+\frac{c (1-c x) d}{c x+1}+e-\frac{e (1-c x)}{c x+1}}{\frac{(1-c x) c}{c x+1}+c}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*a)/(e*Sqrt[d + e*x]) - (2*b*ArcSech[c*x])/(e*Sqrt[d + e*x]) + ((4*I)*b*(2*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqr
t[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[
c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 -
c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1
- c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I
+ Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt
[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] + Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c
*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)
]))]*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e
)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] + (2*I)*Sqrt[((-I)*(Sqrt[-(c*d) - e]*S
qrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqr
t[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1
- c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(
1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*(EllipticPi[(I*Sqrt[-(c*d) - e] - Sqrt[c*d - e])/(Sqrt[-(c*d)
- e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/(
(Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])
^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - EllipticPi[((-I)*Sqrt[-(c*d) - e] + Sqrt[c*d - e])/(Sqrt[-(c*d) -
e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((S
qrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2
/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2])))/(e*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/
(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x)
)*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))])

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Maple [B]  time = 0.271, size = 253, normalized size = 2.4 \begin{align*} 2\,{\frac{1}{e} \left ( -{\frac{a}{\sqrt{ex+d}}}+b \left ( -{\frac{{\rm arcsech} \left (cx\right )}{\sqrt{ex+d}}}-2\,{\frac{c{e}^{2}x}{d \left ( \left ( ex+d \right ) ^{2}{c}^{2}-2\, \left ( ex+d \right ){c}^{2}d+{c}^{2}{d}^{2}-{e}^{2} \right ) }\sqrt{-{\frac{ \left ( ex+d \right ) c-cd-e}{cxe}}}\sqrt{{\frac{ \left ( ex+d \right ) c-cd+e}{cxe}}}{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{cd+e}}},{\frac{cd+e}{cd}},{\sqrt{{\frac{c}{cd-e}}}{\frac{1}{\sqrt{{\frac{c}{cd+e}}}}}} \right ) \sqrt{-{\frac{ \left ( ex+d \right ) c-cd+e}{cd-e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-cd-e}{cd+e}}}{\frac{1}{\sqrt{{\frac{c}{cd+e}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*(-a/(e*x+d)^(1/2)+b*(-1/(e*x+d)^(1/2)*arcsech(c*x)-2*c*e^2*(-((e*x+d)*c-c*d-e)/c/x/e)^(1/2)*x*(((e*x+d)*c-
c*d+e)/c/x/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)*c-c*d+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-c*d-e)/(c*d+e))^(1/2)/d/(c/(c*d+e))^(1/2)/((e*x+d)^2*c^2-2
*(e*x+d)*c^2*d+c^2*d^2-e^2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asech}{\left (c x \right )}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))/(d + e*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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