3.1915 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx\)

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)*Ar
cTanh[(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 = 0.269974, 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 \[ \frac{c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{5/2}}-\frac{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x)}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^3} \]

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)*Ar
cTanh[(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 in Sympy [A]  time = 47.6934, size = 163, normalized size = 0.96 \[ \frac{c^{\frac{3}{2}} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{e^{\frac{5}{2}}} - \frac{2 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{2} \left (d + e x\right )} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**(3/2)*d**(3/2)*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)
*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/e**(5/2) - 2*c*d*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))/(e**2*(d + e*x)) - 2*(a*d*e + c*d*e*x**2 + x*(
a*e**2 + c*d**2))**(3/2)/(3*e*(d + e*x)**3)

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Mathematica [A]  time = 0.3652, size = 146, normalized size = 0.86 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{3 c^{3/2} d^{3/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{\sqrt{d+e x} \sqrt{a e+c d x}}-\frac{2 \sqrt{e} \left (a e^2+c d (3 d+4 e x)\right )}{(d+e x)^2}\right )}{3 e^{5/2}} \]

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]

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

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Maple [B]  time = 0.017, size = 914, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(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*(d/e+x)^2*e+(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*(d/e+x)^2*e+(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*(d/e+x)^2*e+(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*(d/e+x)^2*e+(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*(d/e+x)^2*e+(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*(d/e+x)^2*e+(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+c*d*e*(d/e+x))/(d
*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(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+c*d*e*(d/e+x))/(d*e*c)^(1/2)+
(c*d*(d/e+x)^2*e+(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+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(
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*(d/e
+x)^2*e+(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*(d/e+x
)^2*e+(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+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1
/2))/(d*e*c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.419338, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt{\frac{c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \,{\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{c d}{e}} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{6 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac{3 \,{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt{-\frac{c d}{e}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-\frac{c d}{e}} e}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(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*(2*c*d*e*x + c*d^2 + a*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a
*e^2)*x)*sqrt(-c*d/e)*e)) - 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. \[ \text{Timed out} \]

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError