3.2042 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=236 \[ -\frac{15 c^2 d^2 \sqrt{c d^2-a e^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{7/2}}+\frac{15 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^3 \sqrt{d+e x}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e^2 (d+e x)^{5/2}} \]

[Out]

(15*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e^3*Sqrt[d + e*x]) -
(5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(4*e^2*(d + e*x)^(5/2)) -
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(2*e*(d + e*x)^(9/2)) - (15*c^2*d
^2*Sqrt[c*d^2 - a*e^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])])/(4*e^(7/2))

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Rubi [A]  time = 0.485766, antiderivative size = 236, 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 \[ -\frac{15 c^2 d^2 \sqrt{c d^2-a e^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{7/2}}+\frac{15 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^3 \sqrt{d+e x}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e^2 (d+e x)^{5/2}} \]

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)^(11/2),x]

[Out]

(15*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e^3*Sqrt[d + e*x]) -
(5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(4*e^2*(d + e*x)^(5/2)) -
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(2*e*(d + e*x)^(9/2)) - (15*c^2*d
^2*Sqrt[c*d^2 - a*e^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])])/(4*e^(7/2))

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Rubi in Sympy [A]  time = 99.7184, size = 224, normalized size = 0.95 \[ \frac{15 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e^{3} \sqrt{d + e x}} - \frac{15 c^{2} d^{2} \sqrt{a e^{2} - c d^{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{4 e^{\frac{7}{2}}} - \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{2 e \left (d + e x\right )^{\frac{9}{2}}} \]

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)**(5/2)/(e*x+d)**(11/2),x)

[Out]

15*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*e**3*sqrt(d + e*x
)) - 15*c**2*d**2*sqrt(a*e**2 - c*d**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)))/(4*e**(7/2)) - 5*c*d
*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(4*e**2*(d + e*x)**(5/2)) - (
a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(2*e*(d + e*x)**(9/2))

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Mathematica [A]  time = 0.391848, size = 184, normalized size = 0.78 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\sqrt{e} \sqrt{a e+c d x} \left (-2 a^2 e^4-a c d e^2 (5 d+9 e x)+c^2 d^2 \left (15 d^2+25 d e x+8 e^2 x^2\right )\right )-15 c^2 d^2 (d+e x)^2 \sqrt{a e^2-c d^2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )\right )}{4 e^{7/2} (d+e x)^{5/2} \sqrt{a e+c d x}} \]

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)^(11/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(-2*a^2*e^4 - a*c*d*e^
2*(5*d + 9*e*x) + c^2*d^2*(15*d^2 + 25*d*e*x + 8*e^2*x^2)) - 15*c^2*d^2*Sqrt[-(c
*d^2) + a*e^2]*(d + e*x)^2*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a
*e^2]]))/(4*e^(7/2)*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2))

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Maple [B]  time = 0.04, size = 527, normalized size = 2.2 \[ -{\frac{1}{4\,{e}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}a{c}^{2}{d}^{2}{e}^{4}-15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{3}{d}^{4}{e}^{2}+30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) xa{c}^{2}{d}^{3}{e}^{3}-30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{3}{d}^{5}e+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{c}^{2}{d}^{4}{e}^{2}-15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-8\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+9\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-25\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}+5\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}-15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(11/2),x)

[Out]

-1/4*(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^2*a*c^2*d^2*e^4-15*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+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^3-30*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*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^6-8*x^2*c^2*d^2*
e^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+9*x*a*c*d*e^3*(c*d*x+a*e)^(1/2)*((
a*e^2-c*d^2)*e)^(1/2)-25*x*c^2*d^3*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2
*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4+5*((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)^(5/2)/(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)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29692, size = 1, normalized size = 0. \[ \left [\frac{16 \, c^{3} d^{3} e^{2} x^{3} + 30 \, a c^{2} d^{4} e - 10 \, a^{2} c d^{2} e^{3} - 4 \, a^{3} e^{5} + 15 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{-\frac{c d^{2} - a e^{2}}{e}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e \sqrt{-\frac{c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (25 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \,{\left (15 \, c^{3} d^{5} + 20 \, a c^{2} d^{3} e^{2} - 11 \, a^{2} c d e^{4}\right )} x}{8 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}, \frac{8 \, c^{3} d^{3} e^{2} x^{3} + 15 \, a c^{2} d^{4} e - 5 \, a^{2} c d^{2} e^{3} - 2 \, a^{3} e^{5} - 15 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{\frac{c d^{2} - a e^{2}}{e}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{e}}}\right ) +{\left (25 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (15 \, c^{3} d^{5} + 20 \, a c^{2} d^{3} e^{2} - 11 \, a^{2} c d e^{4}\right )} x}{4 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}\right ] \]

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)^(11/2),x, algorithm="fricas")

[Out]

[1/8*(16*c^3*d^3*e^2*x^3 + 30*a*c^2*d^4*e - 10*a^2*c*d^2*e^3 - 4*a^3*e^5 + 15*(c
^2*d^2*e*x + c^2*d^3)*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)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*s
qrt(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)) + 2*(25*c^3*d^4*e - a*c^2*d^2*e^3)*x^2 + 2*(15*c
^3*d^5 + 20*a*c^2*d^3*e^2 - 11*a^2*c*d*e^4)*x)/((e^4*x + d*e^3)*sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)), 1/4*(8*c^3*d^3*e^2*x^3 + 15*a*c^2*d^
4*e - 5*a^2*c*d^2*e^3 - 2*a^3*e^5 - 15*(c^2*d^2*e*x + c^2*d^3)*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))) + (25*c^3*d^4*e - a*
c^2*d^2*e^3)*x^2 + (15*c^3*d^5 + 20*a*c^2*d^3*e^2 - 11*a^2*c*d*e^4)*x)/((e^4*x +
d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))]

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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)**(11/2),x)

[Out]

Timed out

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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)^(5/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

Timed out