3.1944 \(\int \frac{(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=241 \[ \frac{15 \sqrt{e} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2}}+\frac{15 e \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^3 d^3}+\frac{5 e (d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c^2 d^2}-\frac{2 (d+e x)^3}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*(d + e*x)^3)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (15*e*(c*d^
2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^3*d^3) + (5*e*(d +
e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*c^2*d^2) + (15*Sqrt[e]*(c*d
^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c^(7/2)*d^(7/2))

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Rubi [A]  time = 0.476151, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{15 \sqrt{e} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2}}+\frac{15 e \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^3 d^3}+\frac{5 e (d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c^2 d^2}-\frac{2 (d+e x)^3}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^3)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (15*e*(c*d^
2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^3*d^3) + (5*e*(d +
e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*c^2*d^2) + (15*Sqrt[e]*(c*d
^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c^(7/2)*d^(7/2))

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Rubi in Sympy [A]  time = 71.8283, size = 233, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )^{3}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{5 e \left (d + e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 c^{2} d^{2}} - \frac{15 e \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 c^{3} d^{3}} + \frac{15 \sqrt{e} \left (a e^{2} - c d^{2}\right )^{2} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 c^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(d + e*x)**3/(c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 5*e*(d +
e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(2*c**2*d**2) - 15*e*(a*e**2
- c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*c**3*d**3) + 15*sqr
t(e)*(a*e**2 - c*d**2)**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))))/(8*c**(7/2)*d**(7/2))

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Mathematica [A]  time = 0.419814, size = 196, normalized size = 0.81 \[ \frac{15 \sqrt{e} \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )-2 \sqrt{c} \sqrt{d} (d+e x) \left (15 a^2 e^4+5 a c d e^2 (e x-5 d)+c^2 d^2 \left (8 d^2-9 d e x-2 e^2 x^2\right )\right )}{8 c^{7/2} d^{7/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(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)*(15*a^2*e^4 + 5*a*c*d*e^2*(-5*d + e*x) + c^2*d^2*(
8*d^2 - 9*d*e*x - 2*e^2*x^2)) + 15*Sqrt[e]*(c*d^2 - a*e^2)^2*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*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)])/(8*c^(7/2)*d^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.015, size = 1428, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/4*e^5*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)*x*a^2+15/8*e^9/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)*x*a^4+15/16*e^6/d^4/c^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*a^3+15/8*e*d/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)-49/16*d^6*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)+11/4*e^2/c*x^2/(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)+73/16*e^2/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+2*
d^4*(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)-55/16*e^4/d^2/c^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^
2+1/2*e^3*x^3/d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-11/8*e^6/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^3-25/16*e^2*d^
4/(-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-15/
8*e*d/c*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+21/8*e^4*d^2/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*e^3*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)*x*a-49/8*e*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)*x+15/
16*e^10/d^4/c^4/(-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^5-25/16*e^8/d^2/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+15/4*e^3/d/c^2*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*a-15/4*e^3/d/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+15/8*e^5/d^3/c^3*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^2-15/8*e^5/d^3/c^3*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/4*e^4/
d^2/c^2*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-5*e^7/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)*x*a^3-49/16*d^2/c/(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(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((e*x + d)^4/(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 = 0.563123, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 8 \, c^{2} d^{4} + 25 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} +{\left (9 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{16 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}, \frac{15 \,{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) + 2 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 8 \, c^{2} d^{4} + 25 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} +{\left (9 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{8 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(15*(a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5 + (c^3*d^5 - 2*a*c^2*d^3*e^2
+ a^2*c*d*e^4)*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*(2*c^2*d^2*e^2*
x^2 - 8*c^2*d^4 + 25*a*c*d^2*e^2 - 15*a^2*e^4 + (9*c^2*d^3*e - 5*a*c*d*e^3)*x)*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*x + a*c^3*d^3*e), 1/8*(15*(
a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5 + (c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e
^4)*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*(2*c^2*d^2*e^2*x^2 - 8*c^2*d^
4 + 25*a*c*d^2*e^2 - 15*a^2*e^4 + (9*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*x + a*c^3*d^3*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263711, size = 648, normalized size = 2.69 \[ \frac{{\left ({\left (\frac{2 \,{\left (c^{4} d^{6} e^{5} - 2 \, a c^{3} d^{4} e^{7} + a^{2} c^{2} d^{2} e^{9}\right )} x}{c^{5} d^{7} e^{2} - 2 \, a c^{4} d^{5} e^{4} + a^{2} c^{3} d^{3} e^{6}} + \frac{11 \, c^{4} d^{7} e^{4} - 27 \, a c^{3} d^{5} e^{6} + 21 \, a^{2} c^{2} d^{3} e^{8} - 5 \, a^{3} c d e^{10}}{c^{5} d^{7} e^{2} - 2 \, a c^{4} d^{5} e^{4} + a^{2} c^{3} d^{3} e^{6}}\right )} x + \frac{c^{4} d^{8} e^{3} + 18 \, a c^{3} d^{6} e^{5} - 54 \, a^{2} c^{2} d^{4} e^{7} + 50 \, a^{3} c d^{2} e^{9} - 15 \, a^{4} e^{11}}{c^{5} d^{7} e^{2} - 2 \, a c^{4} d^{5} e^{4} + a^{2} c^{3} d^{3} e^{6}}\right )} x - \frac{8 \, c^{4} d^{9} e^{2} - 41 \, a c^{3} d^{7} e^{4} + 73 \, a^{2} c^{2} d^{5} e^{6} - 55 \, a^{3} c d^{3} e^{8} + 15 \, a^{4} d e^{10}}{c^{5} d^{7} e^{2} - 2 \, a c^{4} d^{5} e^{4} + a^{2} c^{3} d^{3} e^{6}}}{4 \, \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{15 \,{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{8 \, c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(((2*(c^4*d^6*e^5 - 2*a*c^3*d^4*e^7 + a^2*c^2*d^2*e^9)*x/(c^5*d^7*e^2 - 2*a*
c^4*d^5*e^4 + a^2*c^3*d^3*e^6) + (11*c^4*d^7*e^4 - 27*a*c^3*d^5*e^6 + 21*a^2*c^2
*d^3*e^8 - 5*a^3*c*d*e^10)/(c^5*d^7*e^2 - 2*a*c^4*d^5*e^4 + a^2*c^3*d^3*e^6))*x
+ (c^4*d^8*e^3 + 18*a*c^3*d^6*e^5 - 54*a^2*c^2*d^4*e^7 + 50*a^3*c*d^2*e^9 - 15*a
^4*e^11)/(c^5*d^7*e^2 - 2*a*c^4*d^5*e^4 + a^2*c^3*d^3*e^6))*x - (8*c^4*d^9*e^2 -
41*a*c^3*d^7*e^4 + 73*a^2*c^2*d^5*e^6 - 55*a^3*c*d^3*e^8 + 15*a^4*d*e^10)/(c^5*
d^7*e^2 - 2*a*c^4*d^5*e^4 + a^2*c^3*d^3*e^6))/sqrt(c*d*x^2*e + a*d*e + (c*d^2 +
a*e^2)*x) - 15/8*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*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^4*d^4)