3.2041 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{5 c d \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e \sqrt{d+e x}}+\frac{5 c d \left (c d^2-a e^2\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{7/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.522664, antiderivative size = 233, 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 \[ -\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^{7/2}}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{5 c d \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e \sqrt{d+e x}}+\frac{5 c d \left (c d^2-a e^2\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 98.0608, size = 223, normalized size = 0.96 \[ \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{5 c d \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{3} \sqrt{d + e x}} - \frac{5 c d \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{e^{\frac{7}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(5/2)/(e*x+d)**(9/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.52703, size = 169, normalized size = 0.73 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{-\frac{3 \left (c d^2-a e^2\right )^2}{d+e x}+2 c d \left (7 a e^2-6 c d^2\right )+2 c^2 d^2 e x}{3 e^3 (a e+c d x)^2}-\frac{5 c d \left (a e^2-c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{e^{7/2} (a e+c d x)^{5/2}}\right )}{(d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((2*c*d*(-6*c*d^2 + 7*a*e^2) + 2*c^2*d^2*e*x -
(3*(c*d^2 - a*e^2)^2)/(d + e*x))/(3*e^3*(a*e + c*d*x)^2) - (5*c*d*(-(c*d^2) + a*
e^2)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]])/(e^(7/2)
*(a*e + c*d*x)^(5/2))))/(d + e*x)^(5/2)

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 521, normalized size = 2.2 \[ -{\frac{1}{3\,{e}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{a}^{2}cd{e}^{5}-30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) xa{c}^{2}{d}^{3}{e}^{3}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{3}{d}^{5}e+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){a}^{2}c{d}^{2}{e}^{4}-30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{c}^{2}{d}^{4}{e}^{2}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-2\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-14\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+10\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}-20\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}+15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(5/2)/(e*x+d)^(9/2),x)

[Out]

-1/3*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/((a
*e^2-c*d^2)*e)^(1/2))*x*a^2*c*d*e^5-30*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2
)*e)^(1/2))*x*a*c^2*d^3*e^3+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/
2))*x*c^3*d^5*e+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a^2*c*d^
2*e^4-30*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^4*e^2+15*a
rctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^6-2*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)+10*x*c^2*d^3*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+3*((a*
e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4-20*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+
a*e)^(1/2)*a*c*d^2*e^2+15*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^4)/(e*
x+d)^(3/2)/(c*d*x+a*e)^(1/2)/e^3/((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)^(5/2)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.237507, size = 1, normalized size = 0. \[ \left [\frac{4 \, c^{3} d^{3} e^{2} x^{3} - 30 \, a c^{2} d^{4} e + 40 \, a^{2} c d^{2} e^{3} - 6 \, a^{3} e^{5} + 15 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{-\frac{c d^{2} - a e^{2}}{e}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e \sqrt{-\frac{c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \,{\left (5 \, c^{3} d^{4} e - 8 \, a c^{2} d^{2} e^{3}\right )} x^{2} - 2 \,{\left (15 \, c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} - 11 \, a^{2} c d e^{4}\right )} x}{6 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e^{3}}, \frac{2 \, c^{3} d^{3} e^{2} x^{3} - 15 \, a c^{2} d^{4} e + 20 \, a^{2} c d^{2} e^{3} - 3 \, a^{3} e^{5} + 15 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{\frac{c d^{2} - a e^{2}}{e}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{e}}}\right ) - 2 \,{\left (5 \, c^{3} d^{4} e - 8 \, a c^{2} d^{2} e^{3}\right )} x^{2} -{\left (15 \, c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} - 11 \, a^{2} c d e^{4}\right )} x}{3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e^{3}}\right ] \]

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)^(5/2)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

[1/6*(4*c^3*d^3*e^2*x^3 - 30*a*c^2*d^4*e + 40*a^2*c*d^2*e^3 - 6*a^3*e^5 + 15*(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*x + d)*sqr
t(-(c*d^2 - a*e^2)/e)*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)*sqrt(e*x + d)*e*sqrt(-(c*d^2 - a*e^2)/e)
)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(5*c^3*d^4*e - 8*a*c^2*d^2*e^3)*x^2 - 2*(15*c^3
*d^5 - 10*a*c^2*d^3*e^2 - 11*a^2*c*d*e^4)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(e*x + d)*e^3), 1/3*(2*c^3*d^3*e^2*x^3 - 15*a*c^2*d^4*e + 20*a^2*c
*d^2*e^3 - 3*a^3*e^5 + 15*(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*x + d)*sqrt((c*d^2 - a*e^2)/e)*arctan(-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)/((c*d*e^2*x^2 + a*d*e^2 + (c
*d^2*e + a*e^3)*x)*sqrt((c*d^2 - a*e^2)/e))) - 2*(5*c^3*d^4*e - 8*a*c^2*d^2*e^3)
*x^2 - (15*c^3*d^5 - 10*a*c^2*d^3*e^2 - 11*a^2*c*d*e^4)*x)/(sqrt(c*d*e*x^2 + a*d
*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*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)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

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)^(5/2)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError