3.2033 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=250 \[ \frac{c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 e^{5/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^2 (d+e x)^{5/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]

[Out]

-(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*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^2*(c*d^2 - a*e^2)*(d + e
*x)^(3/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*e*(d + e*x)^(9/2))
+ (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^(5/2)*(c*d^2 - a*e^2)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.470568, antiderivative size = 250, 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 \[ \frac{c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 e^{5/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^2 (d+e x)^{5/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]

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)^(11/2),x]

[Out]

-(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*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^2*(c*d^2 - a*e^2)*(d + e
*x)^(3/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*e*(d + e*x)^(9/2))
+ (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^(5/2)*(c*d^2 - a*e^2)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 101.816, size = 230, normalized size = 0.92 \[ \frac{c^{3} d^{3} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{8 e^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} - \frac{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{9}{2}}} \]

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)**(3/2)/(e*x+d)**(11/2),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**(5/2)*(a*e**2 - c*d**2)**(3/2)) - c**2*d**2*
sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*e**2*(d + e*x)**(3/2)*(a*e**2
- c*d**2)) - c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*e**2*(d + e*x
)**(5/2)) - (a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*e*(d + e*x)**(9
/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.505721, size = 194, normalized size = 0.78 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{-8 a^2 e^4+2 a c d e^2 (d-7 e x)+c^2 d^2 \left (3 d^2+8 d e x-3 e^2 x^2\right )}{3 e^2 (d+e x)^3 \left (a e^2-c d^2\right ) (a e+c d x)}+\frac{c^3 d^3 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{e^{5/2} \left (a e^2-c d^2\right )^{3/2} (a e+c d x)^{3/2}}\right )}{8 (d+e x)^{3/2}} \]

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)^(11/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.038, size = 457, normalized size = 1.8 \[{\frac{1}{ \left ( 24\,a{e}^{2}-24\,c{d}^{2} \right ){e}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{3}{c}^{3}{d}^{3}{e}^{3}+9\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{3}{d}^{4}{e}^{2}+9\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{3}{d}^{5}e+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-3\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-14\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+8\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(11/2),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)-14*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+2*((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)/(c*d*x+a*e)^(1/2)/(a*e^2-c*d^2)/e^2/((a*e^2-c*d^2)*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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238642, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} - 2 \,{\left (4 \, c^{2} d^{3} e - 7 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d^{2} e + a e^{3}} \sqrt{e x + d} + 3 \,{\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \log \left (-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d} +{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{48 \,{\left (c d^{6} e^{2} - a d^{4} e^{4} +{\left (c d^{2} e^{6} - a e^{8}\right )} x^{4} + 4 \,{\left (c d^{3} e^{5} - a d e^{7}\right )} x^{3} + 6 \,{\left (c d^{4} e^{4} - a d^{2} e^{6}\right )} x^{2} + 4 \,{\left (c d^{5} e^{3} - a d^{3} e^{5}\right )} x\right )} \sqrt{-c d^{2} e + a e^{3}}}, \frac{{\left (3 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} - 2 \,{\left (4 \, c^{2} d^{3} e - 7 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d} - 3 \,{\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right )}{24 \,{\left (c d^{6} e^{2} - a d^{4} e^{4} +{\left (c d^{2} e^{6} - a e^{8}\right )} x^{4} + 4 \,{\left (c d^{3} e^{5} - a d e^{7}\right )} x^{3} + 6 \,{\left (c d^{4} e^{4} - a d^{2} e^{6}\right )} x^{2} + 4 \,{\left (c d^{5} e^{3} - a d^{3} e^{5}\right )} x\right )} \sqrt{c d^{2} e - a e^{3}}}\right ] \]

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)^(11/2),x, algorithm="fricas")

[Out]

[1/48*(2*(3*c^2*d^2*e^2*x^2 - 3*c^2*d^4 - 2*a*c*d^2*e^2 + 8*a^2*e^4 - 2*(4*c^2*d
^3*e - 7*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*d^6*e^2 - a*d^4*e^4
+ (c*d^2*e^6 - a*e^8)*x^4 + 4*(c*d^3*e^5 - a*d*e^7)*x^3 + 6*(c*d^4*e^4 - a*d^2*e
^6)*x^2 + 4*(c*d^5*e^3 - a*d^3*e^5)*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 - 2*a*c*d^2*e^2 + 8*a^2*e^4 - 2*(4*c^2*d^3*e - 7*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)*sqrt(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*d^6*e^2 -
a*d^4*e^4 + (c*d^2*e^6 - a*e^8)*x^4 + 4*(c*d^3*e^5 - a*d*e^7)*x^3 + 6*(c*d^4*e^
4 - a*d^2*e^6)*x^2 + 4*(c*d^5*e^3 - a*d^3*e^5)*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)**(3/2)/(e*x+d)**(11/2),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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

Timed out