3.2014 \(\int \frac{(d+e x)^{3/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{3/2}}-\frac{\sqrt{d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \]

[Out]

-(Sqrt[d + e*x]/((c*d^2 - a*e^2)*(a*e + c*d*x))) + (e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e
^2]])/(Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(3/2))

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Rubi [A]  time = 0.0565612, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {626, 51, 63, 208} \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{3/2}}-\frac{\sqrt{d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]/((c*d^2 - a*e^2)*(a*e + c*d*x))) + (e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e
^2]])/(Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(3/2))

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0770845, size = 101, normalized size = 1. \[ \frac{e \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{a e^2-c d^2}}\right )}{\sqrt{c} \sqrt{d} \left (a e^2-c d^2\right )^{3/2}}-\frac{\sqrt{d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]/((c*d^2 - a*e^2)*(a*e + c*d*x))) + (e*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a
*e^2]])/(Sqrt[c]*Sqrt[d]*(-(c*d^2) + a*e^2)^(3/2))

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Maple [A]  time = 0.193, size = 99, normalized size = 1. \begin{align*}{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{e}{a{e}^{2}-c{d}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84347, size = 725, normalized size = 7.18 \begin{align*} \left [-\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + a e^{2}\right )} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{c^{2} d^{3} - a c d e^{2}} \sqrt{e x + d}}{c d x + a e}\right ) + 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{2 \,{\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x\right )}}, -\frac{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (c d e x + a e^{2}\right )} \arctan \left (\frac{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}{c d e x + c d^{2}}\right ) +{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(c^2*d^3 - a*c*d*e^2)*(c*d*e*x + a*e^2)*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(c^2*d^3 - a*c*d*e^2
)*sqrt(e*x + d))/(c*d*x + a*e)) + 2*(c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(a*c^3*d^5*e - 2*a^2*c^2*d^3*e^3 + a^
3*c*d*e^5 + (c^4*d^6 - 2*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x), -(sqrt(-c^2*d^3 + a*c*d*e^2)*(c*d*e*x + a*e^2)*a
rctan(sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d)/(c*d*e*x + c*d^2)) + (c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(a*c^
3*d^5*e - 2*a^2*c^2*d^3*e^3 + a^3*c*d*e^5 + (c^4*d^6 - 2*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out