3.2040 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=240 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac{2 \left (a-\frac{c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}-\frac{2 \left (c d^2-a e^2\right )^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{7/2}}+\frac{2 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.60168, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac{2 \left (a-\frac{c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}-\frac{2 \left (c d^2-a e^2\right )^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{7/2}}+\frac{2 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 \sqrt{d+e x}} \]

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)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 102.91, size = 228, normalized size = 0.95 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} + \frac{2 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{2 \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{3} \sqrt{d + e x}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{e^{\frac{7}{2}}} \]

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)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.392334, size = 170, normalized size = 0.71 \[ \frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \left (\sqrt{e} \sqrt{a e+c d x} \left (23 a^2 e^4+a c d e^2 (11 e x-35 d)+c^2 d^2 \left (15 d^2-5 d e x+3 e^2 x^2\right )\right )-15 \left (a e^2-c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )\right )}{15 e^{7/2} \sqrt{(d+e x) (a e+c d x)}} \]

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)^(7/2),x]

[Out]

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(23*a^2*e^4 + a*c*
d*e^2*(-35*d + 11*e*x) + c^2*d^2*(15*d^2 - 5*d*e*x + 3*e^2*x^2)) - 15*(-(c*d^2)
+ a*e^2)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]]))/(15
*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.026, size = 437, normalized size = 1.8 \[ -{\frac{2}{15\,{e}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){a}^{3}{e}^{6}-45\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){a}^{2}c{d}^{2}{e}^{4}+45\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{c}^{2}{d}^{4}{e}^{2}-15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-3\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-11\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+5\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-23\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}+35\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}-15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(7/2),x)

[Out]

-2/15*(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))*a^3*e^6-45*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+45*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*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)-11*x*a*c*d*e^3*(c*d*x+a*e
)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+5*x*c^2*d^3*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e
)^(1/2)-23*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4+35*((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)^(1/2)/(c*d*x+a*e)^(1/2)/e^3/((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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230807, size = 1, normalized size = 0. \[ \left [\frac{6 \, c^{3} d^{3} e^{3} x^{4} + 30 \, a c^{2} d^{5} e - 70 \, a^{2} c d^{3} e^{3} + 46 \, a^{3} d e^{5} - 4 \,{\left (c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{3} + 15 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{-\frac{c d^{2} - a e^{2}}{e}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e \sqrt{-\frac{c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 4 \,{\left (5 \, c^{3} d^{5} e - 13 \, a c^{2} d^{3} e^{3} + 17 \, a^{2} c d e^{5}\right )} x^{2} + 2 \,{\left (15 \, c^{3} d^{6} - 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + 23 \, a^{3} e^{6}\right )} x}{15 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e^{3}}, \frac{2 \,{\left (3 \, c^{3} d^{3} e^{3} x^{4} + 15 \, a c^{2} d^{5} e - 35 \, a^{2} c d^{3} e^{3} + 23 \, a^{3} d e^{5} - 2 \,{\left (c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{3} - 15 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{\frac{c d^{2} - a e^{2}}{e}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{e}}}\right ) + 2 \,{\left (5 \, c^{3} d^{5} e - 13 \, a c^{2} d^{3} e^{3} + 17 \, a^{2} c d e^{5}\right )} x^{2} +{\left (15 \, c^{3} d^{6} - 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + 23 \, a^{3} e^{6}\right )} x\right )}}{15 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e^{3}}\right ] \]

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)^(7/2),x, algorithm="fricas")

[Out]

[1/15*(6*c^3*d^3*e^3*x^4 + 30*a*c^2*d^5*e - 70*a^2*c*d^3*e^3 + 46*a^3*d*e^5 - 4*
(c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^3 + 15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqr
t(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)*
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^5*e - 13*a*c^2*d^3*e^3 + 17*a^2*c*d*e^5)*x^2 + 2*(15*c^3*d^6
- 25*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 + 23*a^3*e^6)*x)/(sqrt(c*d*e*x^2 + a*d*e + (
c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e^3), 2/15*(3*c^3*d^3*e^3*x^4 + 15*a*c^2*d^5*e -
35*a^2*c*d^3*e^3 + 23*a^3*d*e^5 - 2*(c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^3 - 15*(c
^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(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^5*e - 13*a*c^2*d^3*e^3 + 17*a^2*c*d*
e^5)*x^2 + (15*c^3*d^6 - 25*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 + 23*a^3*e^6)*x)/(sqrt
(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e^3)]

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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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)^(7/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError