3.2024 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=264 \[ \frac{c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*e*(d + e*x)^(7/2)) + (c*d*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) +
(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e*(c*d^2 - a*e^2)^2*(d
+ e*x)^(3/2)) + (c^3*d^3*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])])/(8*e^(3/2)*(c*d^2 - a*e^2)^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.524692, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*e*(d + e*x)^(7/2)) + (c*d*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) +
(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e*(c*d^2 - a*e^2)^2*(d
+ e*x)^(3/2)) + (c^3*d^3*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])])/(8*e^(3/2)*(c*d^2 - a*e^2)^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 106.261, size = 238, normalized size = 0.9 \[ - \frac{c^{3} d^{3} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{8 e^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 e \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{12 e \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(1/2)/(e*x+d)**(9/2),x)

[Out]

-c**3*d**3*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)))/(8*e**(3/2)*(a*e**2 - c*d**2)**(5/2)) + c**2*d**2
*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*e*(d + e*x)**(3/2)*(a*e**2 -
c*d**2)**2) - c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(12*e*(d + e*x)
**(5/2)*(a*e**2 - c*d**2)) - sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3*e
*(d + e*x)**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.732789, size = 181, normalized size = 0.69 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\frac{3 c^2 d^2 (d+e x)^2}{e \left (c d^2-a e^2\right )^2}+\frac{2 c d (d+e x)}{c d^2 e-a e^3}-\frac{8}{e}}{3 (d+e x)^3}-\frac{c^3 d^3 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{e^{3/2} \left (a e^2-c d^2\right )^{5/2} \sqrt{a e+c d x}}\right )}{8 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.038, size = 457, normalized size = 1.7 \[ -{\frac{1}{24\,e \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{3}{c}^{3}{d}^{3}{e}^{3}+9\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{3}{d}^{4}{e}^{2}+9\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{3}{d}^{5}e+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-3\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+2\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-8\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}-14\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}{\frac{1}{\sqrt{cdx+ae}}}} \]

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)^(1/2)/(e*x+d)^(9/2),x)

[Out]

-1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a
*e^2-c*d^2)*e)^(1/2))*x^3*c^3*d^3*e^3+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^
2)*e)^(1/2))*x^2*c^3*d^4*e^2+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/
2))*x*c^3*d^5*e+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^6-3
*x^2*c^2*d^2*e^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*x*a*c*d*e^3*(c*d*x+
a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-8*x*c^2*d^3*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2
)*e)^(1/2)+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4-14*((a*e^2-c*d^2)
*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e^2+3*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1
/2)*c^2*d^4)/(e*x+d)^(7/2)/((a*e^2-c*d^2)*e)^(1/2)/e/(a*e^2-c*d^2)^2/(c*d*x+a*e)
^(1/2)

_______________________________________________________________________________________

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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.237085, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

[1/48*(2*(3*c^2*d^2*e^2*x^2 - 3*c^2*d^4 + 14*a*c*d^2*e^2 - 8*a^2*e^4 + 2*(4*c^2*
d^3*e - a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d^2*e
+ a*e^3)*sqrt(e*x + d) + 3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*
x^2 + 4*c^3*d^6*e*x + c^3*d^7)*log(-(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)*(c*d^2*e - a*e^3)*sqrt(e*x + d) + (c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^
2)*sqrt(-c*d^2*e + a*e^3))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d^8*e - 2*a*c*d^6*e
^3 + a^2*d^4*e^5 + (c^2*d^4*e^5 - 2*a*c*d^2*e^7 + a^2*e^9)*x^4 + 4*(c^2*d^5*e^4
- 2*a*c*d^3*e^6 + a^2*d*e^8)*x^3 + 6*(c^2*d^6*e^3 - 2*a*c*d^4*e^5 + a^2*d^2*e^7)
*x^2 + 4*(c^2*d^7*e^2 - 2*a*c*d^5*e^4 + a^2*d^3*e^6)*x)*sqrt(-c*d^2*e + a*e^3)),
1/24*((3*c^2*d^2*e^2*x^2 - 3*c^2*d^4 + 14*a*c*d^2*e^2 - 8*a^2*e^4 + 2*(4*c^2*d^
3*e - a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a
*e^3)*sqrt(e*x + d) - 3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2
+ 4*c^3*d^6*e*x + c^3*d^7)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*s
qrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)
))/((c^2*d^8*e - 2*a*c*d^6*e^3 + a^2*d^4*e^5 + (c^2*d^4*e^5 - 2*a*c*d^2*e^7 + a^
2*e^9)*x^4 + 4*(c^2*d^5*e^4 - 2*a*c*d^3*e^6 + a^2*d*e^8)*x^3 + 6*(c^2*d^6*e^3 -
2*a*c*d^4*e^5 + a^2*d^2*e^7)*x^2 + 4*(c^2*d^7*e^2 - 2*a*c*d^5*e^4 + a^2*d^3*e^6)
*x)*sqrt(c*d^2*e - a*e^3))]

_______________________________________________________________________________________

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)**(1/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

Timed out