3.2061 \(\int \frac{1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=393 \[ -\frac{315 c^4 d^4 \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{315 c^4 d^4 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{11/2}}+\frac{105 c^3 d^3}{64 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{21 c^2 d^2}{32 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{3 c d}{8 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{4 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]
) + (3*c*d)/(8*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (21*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3)/(64*(c*d^2 - a*e^2)^4*Sqrt[d + e
*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (315*c^4*d^4*Sqrt[d + e*x])/(
64*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (315*c^4*d^4
*Sqrt[e]*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^
2 - a*e^2]*Sqrt[d + e*x])])/(64*(c*d^2 - a*e^2)^(11/2))

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Rubi [A]  time = 0.866836, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{315 c^4 d^4 \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{315 c^4 d^4 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{11/2}}+\frac{105 c^3 d^3}{64 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{21 c^2 d^2}{32 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{3 c d}{8 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{4 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]
) + (3*c*d)/(8*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (21*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3)/(64*(c*d^2 - a*e^2)^4*Sqrt[d + e
*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (315*c^4*d^4*Sqrt[d + e*x])/(
64*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (315*c^4*d^4
*Sqrt[e]*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^
2 - a*e^2]*Sqrt[d + e*x])])/(64*(c*d^2 - a*e^2)^(11/2))

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Rubi in Sympy [A]  time = 174.246, size = 371, normalized size = 0.94 \[ - \frac{315 c^{4} d^{4} \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{64 \left (a e^{2} - c d^{2}\right )^{\frac{11}{2}}} + \frac{315 c^{4} d^{4} \sqrt{d + e x}}{64 \left (a e^{2} - c d^{2}\right )^{5} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{105 c^{3} d^{3}}{64 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{21 c^{2} d^{2}}{32 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{3 c d}{8 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{1}{4 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

-315*c**4*d**4*sqrt(e)*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**
2))/(sqrt(d + e*x)*sqrt(a*e**2 - c*d**2)))/(64*(a*e**2 - c*d**2)**(11/2)) + 315*
c**4*d**4*sqrt(d + e*x)/(64*(a*e**2 - c*d**2)**5*sqrt(a*d*e + c*d*e*x**2 + x*(a*
e**2 + c*d**2))) + 105*c**3*d**3/(64*sqrt(d + e*x)*(a*e**2 - c*d**2)**4*sqrt(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 21*c**2*d**2/(32*(d + e*x)**(3/2)*(a*e
**2 - c*d**2)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 3*c*d/(8*(d +
e*x)**(5/2)*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
) - 1/(4*(d + e*x)**(7/2)*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2)))

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Mathematica [A]  time = 1.69797, size = 246, normalized size = 0.63 \[ \frac{(d+e x)^{3/2} \left (\frac{(a e+c d x)^2 \left (\frac{128 c^4 d^4}{a e+c d x}+\frac{82 c^2 d^2 e \left (c d^2-a e^2\right )}{(d+e x)^2}+\frac{40 c d e \left (c d^2-a e^2\right )^2}{(d+e x)^3}-\frac{16 e \left (a e^2-c d^2\right )^3}{(d+e x)^4}+\frac{187 c^3 d^3 e}{d+e x}\right )}{\left (a e^2-c d^2\right )^5}-\frac{315 c^4 d^4 \sqrt{e} (a e+c d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{11/2}}\right )}{64 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

((d + e*x)^(3/2)*(((a*e + c*d*x)^2*((128*c^4*d^4)/(a*e + c*d*x) - (16*e*(-(c*d^2
) + a*e^2)^3)/(d + e*x)^4 + (40*c*d*e*(c*d^2 - a*e^2)^2)/(d + e*x)^3 + (82*c^2*d
^2*e*(c*d^2 - a*e^2))/(d + e*x)^2 + (187*c^3*d^3*e)/(d + e*x)))/(-(c*d^2) + a*e^
2)^5 - (315*c^4*d^4*Sqrt[e]*(a*e + c*d*x)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*
x])/Sqrt[-(c*d^2) + a*e^2]])/(-(c*d^2) + a*e^2)^(11/2)))/(64*((a*e + c*d*x)*(d +
e*x))^(3/2))

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Maple [B]  time = 0.049, size = 767, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-1/64*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(315*arctanh(e*(c*d*x+a*e)^(1/2)/(
(a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^4*c^4*d^4*e^5+1260*arctanh(e*(c*d*x+
a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^3*c^4*d^5*e^4+1890*arcta
nh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^2*c^4*d^6*e^
3-315*((a*e^2-c*d^2)*e)^(1/2)*x^4*c^4*d^4*e^4+1260*arctanh(e*(c*d*x+a*e)^(1/2)/(
(a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x*c^4*d^7*e^2-105*((a*e^2-c*d^2)*e)^(1
/2)*x^3*a*c^3*d^3*e^5-1155*((a*e^2-c*d^2)*e)^(1/2)*x^3*c^4*d^5*e^3+315*arctanh(e
*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^4*d^8*e+42*((a*e
^2-c*d^2)*e)^(1/2)*x^2*a^2*c^2*d^2*e^6-399*((a*e^2-c*d^2)*e)^(1/2)*x^2*a*c^3*d^4
*e^4-1533*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^4*d^6*e^2-24*((a*e^2-c*d^2)*e)^(1/2)*x*a
^3*c*d*e^7+156*((a*e^2-c*d^2)*e)^(1/2)*x*a^2*c^2*d^3*e^5-555*((a*e^2-c*d^2)*e)^(
1/2)*x*a*c^3*d^5*e^3-837*((a*e^2-c*d^2)*e)^(1/2)*x*c^4*d^7*e+16*((a*e^2-c*d^2)*e
)^(1/2)*a^4*e^8-88*((a*e^2-c*d^2)*e)^(1/2)*a^3*c*d^2*e^6+210*((a*e^2-c*d^2)*e)^(
1/2)*a^2*c^2*d^4*e^4-325*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^6*e^2-128*((a*e^2-c*d^2
)*e)^(1/2)*c^4*d^8)/(e*x+d)^(9/2)/(c*d*x+a*e)/(a*e^2-c*d^2)^5/((a*e^2-c*d^2)*e)^
(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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251061, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/128*(315*(c^5*d^5*e^5*x^6 + a*c^4*d^9*e + (5*c^5*d^6*e^4 + a*c^4*d^4*e^6)*x^5
+ 5*(2*c^5*d^7*e^3 + a*c^4*d^5*e^5)*x^4 + 10*(c^5*d^8*e^2 + a*c^4*d^6*e^4)*x^3
+ 5*(c^5*d^9*e + 2*a*c^4*d^7*e^3)*x^2 + (c^5*d^10 + 5*a*c^4*d^8*e^2)*x)*sqrt(-e/
(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2
- a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(315*c^4*d^4*e^4*x^4 + 128*c^4*d^8 + 3
25*a*c^3*d^6*e^2 - 210*a^2*c^2*d^4*e^4 + 88*a^3*c*d^2*e^6 - 16*a^4*e^8 + 105*(11
*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 21*(73*c^4*d^6*e^2 + 19*a*c^3*d^4*e^4 - 2*a^
2*c^2*d^2*e^6)*x^2 + 3*(279*c^4*d^7*e + 185*a*c^3*d^5*e^3 - 52*a^2*c^2*d^3*e^5 +
8*a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a
*c^5*d^15*e - 5*a^2*c^4*d^13*e^3 + 10*a^3*c^3*d^11*e^5 - 10*a^4*c^2*d^9*e^7 + 5*
a^5*c*d^7*e^9 - a^6*d^5*e^11 + (c^6*d^11*e^5 - 5*a*c^5*d^9*e^7 + 10*a^2*c^4*d^7*
e^9 - 10*a^3*c^3*d^5*e^11 + 5*a^4*c^2*d^3*e^13 - a^5*c*d*e^15)*x^6 + (5*c^6*d^12
*e^4 - 24*a*c^5*d^10*e^6 + 45*a^2*c^4*d^8*e^8 - 40*a^3*c^3*d^6*e^10 + 15*a^4*c^2
*d^4*e^12 - a^6*e^16)*x^5 + 5*(2*c^6*d^13*e^3 - 9*a*c^5*d^11*e^5 + 15*a^2*c^4*d^
9*e^7 - 10*a^3*c^3*d^7*e^9 + 3*a^5*c*d^3*e^13 - a^6*d*e^15)*x^4 + 10*(c^6*d^14*e
^2 - 4*a*c^5*d^12*e^4 + 5*a^2*c^4*d^10*e^6 - 5*a^4*c^2*d^6*e^10 + 4*a^5*c*d^4*e^
12 - a^6*d^2*e^14)*x^3 + 5*(c^6*d^15*e - 3*a*c^5*d^13*e^3 + 10*a^3*c^3*d^9*e^7 -
15*a^4*c^2*d^7*e^9 + 9*a^5*c*d^5*e^11 - 2*a^6*d^3*e^13)*x^2 + (c^6*d^16 - 15*a^
2*c^4*d^12*e^4 + 40*a^3*c^3*d^10*e^6 - 45*a^4*c^2*d^8*e^8 + 24*a^5*c*d^6*e^10 -
5*a^6*d^4*e^12)*x), 1/64*(315*(c^5*d^5*e^5*x^6 + a*c^4*d^9*e + (5*c^5*d^6*e^4 +
a*c^4*d^4*e^6)*x^5 + 5*(2*c^5*d^7*e^3 + a*c^4*d^5*e^5)*x^4 + 10*(c^5*d^8*e^2 + a
*c^4*d^6*e^4)*x^3 + 5*(c^5*d^9*e + 2*a*c^4*d^7*e^3)*x^2 + (c^5*d^10 + 5*a*c^4*d^
8*e^2)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2)))) - (315*c^4*d^4*e^4*x^4 + 128*c^4*d
^8 + 325*a*c^3*d^6*e^2 - 210*a^2*c^2*d^4*e^4 + 88*a^3*c*d^2*e^6 - 16*a^4*e^8 + 1
05*(11*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 21*(73*c^4*d^6*e^2 + 19*a*c^3*d^4*e^4
- 2*a^2*c^2*d^2*e^6)*x^2 + 3*(279*c^4*d^7*e + 185*a*c^3*d^5*e^3 - 52*a^2*c^2*d^3
*e^5 + 8*a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x +
d))/(a*c^5*d^15*e - 5*a^2*c^4*d^13*e^3 + 10*a^3*c^3*d^11*e^5 - 10*a^4*c^2*d^9*e^
7 + 5*a^5*c*d^7*e^9 - a^6*d^5*e^11 + (c^6*d^11*e^5 - 5*a*c^5*d^9*e^7 + 10*a^2*c^
4*d^7*e^9 - 10*a^3*c^3*d^5*e^11 + 5*a^4*c^2*d^3*e^13 - a^5*c*d*e^15)*x^6 + (5*c^
6*d^12*e^4 - 24*a*c^5*d^10*e^6 + 45*a^2*c^4*d^8*e^8 - 40*a^3*c^3*d^6*e^10 + 15*a
^4*c^2*d^4*e^12 - a^6*e^16)*x^5 + 5*(2*c^6*d^13*e^3 - 9*a*c^5*d^11*e^5 + 15*a^2*
c^4*d^9*e^7 - 10*a^3*c^3*d^7*e^9 + 3*a^5*c*d^3*e^13 - a^6*d*e^15)*x^4 + 10*(c^6*
d^14*e^2 - 4*a*c^5*d^12*e^4 + 5*a^2*c^4*d^10*e^6 - 5*a^4*c^2*d^6*e^10 + 4*a^5*c*
d^4*e^12 - a^6*d^2*e^14)*x^3 + 5*(c^6*d^15*e - 3*a*c^5*d^13*e^3 + 10*a^3*c^3*d^9
*e^7 - 15*a^4*c^2*d^7*e^9 + 9*a^5*c*d^5*e^11 - 2*a^6*d^3*e^13)*x^2 + (c^6*d^16 -
15*a^2*c^4*d^12*e^4 + 40*a^3*c^3*d^10*e^6 - 45*a^4*c^2*d^8*e^8 + 24*a^5*c*d^6*e
^10 - 5*a^6*d^4*e^12)*x)]

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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(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 2]