3.1926 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=169 \[ \frac{c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{5/2}}-\frac{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x)}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^3} \]

[Out]

(-2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^2*(d + e*x)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2))/(3*e*(d + e*x)^3) + (c^(3/2)*d^(3/2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/e^(5/2)

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Rubi [A]  time = 0.0915419, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {662, 621, 206} \[ \frac{c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{5/2}}-\frac{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x)}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

(-2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^2*(d + e*x)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2))/(3*e*(d + e*x)^3) + (c^(3/2)*d^(3/2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/e^(5/2)

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.707244, size = 184, normalized size = 1.09 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (\frac{3 c^{3/2} d^{3/2} \sqrt{c d} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{\sqrt{c d^2-a e^2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{\sqrt{e} \left (a e^2+c d (3 d+4 e x)\right )}{(d+e x)^2}\right )}{3 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(a*e^2 + c*d*(3*d + 4*e*x)))/(d + e*x)^2) + (3*c^(3/2)*d^(3/2)*Sq
rt[c*d]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d^2 - a*
e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(3*e^(5/2))

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Maple [B]  time = 0.049, size = 914, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x)

[Out]

-2/3/e^4/(a*e^2-c*d^2)/(d/e+x)^4*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)-4/3/e^3*d*c/(a*e^2-c*d^2)^2/(d/
e+x)^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)+16/3/e^2*d^2*c^2/(a*e^2-c*d^2)^3/(d/e+x)^2*(c*d*e*(d/e+x)
^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)-16/3/e*d^3*c^3/(a*e^2-c*d^2)^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)-4
*e*d^3*c^3/(a*e^2-c*d^2)^3*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-2*e^2*d^2*c^2/(a*e^2-c*d^2)^3*a^2
*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+e^4*d^2*c^2/(a*e^2-c*d^2)^3*a^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)
*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-3*e^2*d^4*c^3/(a*e^2-c*d^2)
^3*a^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*
e*c)^(1/2)+3*d^6*c^4/(a*e^2-c*d^2)^3*a*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(
a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+4/e*d^5*c^4/(a*e^2-c*d^2)^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))
^(1/2)*x+2/e^2*d^6*c^4/(a*e^2-c*d^2)^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/e^2*d^8*c^5/(a*e^2-c*d^
2)^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*
c)^(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.38952, size = 907, normalized size = 5.37 \begin{align*} \left [\frac{3 \,{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt{\frac{c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \,{\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{c d}{e}} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{6 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac{3 \,{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt{-\frac{c d}{e}} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-\frac{c d}{e}}}{2 \,{\left (c^{2} d^{2} e x^{2} + a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/6*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(c*d/e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*
e^4 + 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e
+ a*c*d*e^3)*x) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + 3*c*d^2 + a*e^2))/(e^4*x^2 + 2*d
*e^3*x + d^2*e^2), -1/3*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(-c*d/e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d/e)/(c^2*d^2*e*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^
2)*x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + 3*c*d^2 + a*e^2))/(e^4*x^2 + 2*d*e^3*x + d
^2*e^2)]

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**4,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError