3.1945 \(\int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=180 \[ \frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2}}+\frac{3 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2}-\frac{2 (d+e x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.292731, antiderivative size = 180, 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 \[ \frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2}}+\frac{3 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2}-\frac{2 (d+e x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.7609, size = 173, normalized size = 0.96 \[ - \frac{2 \left (d + e x\right )^{2}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{3 e \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c^{2} d^{2}} - \frac{3 \sqrt{e} \left (a e^{2} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 c^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.325259, size = 162, normalized size = 0.9 \[ \frac{3 \sqrt{e} \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )-2 \sqrt{c} \sqrt{d} (d+e x) \left (c d (2 d-e x)-3 a e^2\right )}{2 c^{5/2} d^{5/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 1047, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*e^4/d^3/c^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+5/2*e^5/c/(-a^2*e^4
+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-1/2*e^3*d^
2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-9
/2*e*d^4*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*x-3/4*e^8/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*a^4+1/2*e^6/d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+e^4*d/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-3/2*e^3/d^2/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e
*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a+2*e^2
/d/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-5/2*e^2*d^3/(-a^2*e^4+2*a*c*d^2
*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+2*d^3*(2*c*d*e*x+a*e^2+c
*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-9/
4*d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/2*e^3/d^2/c^2*x/(a*e*d+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*a-3/2*e/c*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-9/4*
d^5*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3
/2*e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2))/(d*e*c)^(1/2)+e^2*x^2/d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
-3/2*e^7/d^2/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2)*x*a^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.349505, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{4 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, \frac{3 \,{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{2 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264442, size = 478, normalized size = 2.66 \[ \frac{{\left (\frac{{\left (c^{3} d^{5} e^{3} - 2 \, a c^{2} d^{3} e^{5} + a^{2} c d e^{7}\right )} x}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}} - \frac{c^{3} d^{6} e^{2} - 5 \, a c^{2} d^{4} e^{4} + 7 \, a^{2} c d^{2} e^{6} - 3 \, a^{3} e^{8}}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}}\right )} x - \frac{2 \, c^{3} d^{7} e - 7 \, a c^{2} d^{5} e^{3} + 8 \, a^{2} c d^{3} e^{5} - 3 \, a^{3} d e^{7}}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}}}{\sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{3 \,{\left (c d^{2} e - a e^{3}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2 \, c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(((c^3*d^5*e^3 - 2*a*c^2*d^3*e^5 + a^2*c*d*e^7)*x/(c^4*d^6*e - 2*a*c^3*d^4*e^3 +
a^2*c^2*d^2*e^5) - (c^3*d^6*e^2 - 5*a*c^2*d^4*e^4 + 7*a^2*c*d^2*e^6 - 3*a^3*e^8
)/(c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5))*x - (2*c^3*d^7*e - 7*a*c^2*d^
5*e^3 + 8*a^2*c*d^3*e^5 - 3*a^3*d*e^7)/(c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^
2*e^5))/sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x) - 3/2*(c*d^2*e - a*e^3)*sqrt
(c*d)*e^(-1/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c
*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^3*d^3)