3.2044 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=301 \[ \frac{5 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 e^{7/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{5 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^3 (d+e x)^{5/2}}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}} \]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*e^3*(d + e*x)^(5/2)
) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^3*(c*d^2 - a*e
^2)*(d + e*x)^(3/2)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24
*e^2*(d + e*x)^(9/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(4*e*(d +
e*x)^(13/2)) + (5*c^4*d^4*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])])/(64*e^(7/2)*(c*d^2 - a*e^2)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.623856, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{5 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 e^{7/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{5 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^3 (d+e x)^{5/2}}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}} \]

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)^(15/2),x]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*e^3*(d + e*x)^(5/2)
) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^3*(c*d^2 - a*e
^2)*(d + e*x)^(3/2)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24
*e^2*(d + e*x)^(9/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(4*e*(d +
e*x)^(13/2)) + (5*c^4*d^4*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])])/(64*e^(7/2)*(c*d^2 - a*e^2)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 129.308, size = 286, normalized size = 0.95 \[ \frac{5 c^{4} d^{4} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{64 e^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} - \frac{5 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 e^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )} - \frac{5 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{32 e^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{24 e^{2} \left (d + e x\right )^{\frac{9}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{4 e \left (d + e x\right )^{\frac{13}{2}}} \]

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)**(15/2),x)

[Out]

5*c**4*d**4*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)))/(64*e**(7/2)*(a*e**2 - c*d**2)**(3/2)) - 5*c**3*
d**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(64*e**3*(d + e*x)**(3/2)*(a
*e**2 - c*d**2)) - 5*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3
2*e**3*(d + e*x)**(5/2)) - 5*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/
2)/(24*e**2*(d + e*x)**(9/2)) - (a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2
)/(4*e*(d + e*x)**(13/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.883462, size = 214, normalized size = 0.71 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{5 c^4 d^4 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{e^{7/2} \left (a e^2-c d^2\right )^{3/2} (a e+c d x)^{5/2}}+\frac{\frac{15 c^3 d^3 (d+e x)^3}{c d^2-a e^2}+136 c d (d+e x) \left (c d^2-a e^2\right )-48 \left (c d^2-a e^2\right )^2-118 c^2 d^2 (d+e x)^2}{3 e^3 (d+e x)^4 (a e+c d x)^2}\right )}{64 (d+e x)^{5/2}} \]

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)^(15/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-48*(c*d^2 - a*e^2)^2 + 136*c*d*(c*d^2 - a*e^
2)*(d + e*x) - 118*c^2*d^2*(d + e*x)^2 + (15*c^3*d^3*(d + e*x)^3)/(c*d^2 - a*e^2
))/(3*e^3*(a*e + c*d*x)^2*(d + e*x)^4) + (5*c^4*d^4*ArcTanh[(Sqrt[e]*Sqrt[a*e +
c*d*x])/Sqrt[-(c*d^2) + a*e^2]])/(e^(7/2)*(-(c*d^2) + a*e^2)^(3/2)*(a*e + c*d*x)
^(5/2))))/(64*(d + e*x)^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.041, size = 662, normalized size = 2.2 \[{\frac{1}{192\,{e}^{3} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{4}{c}^{4}{d}^{4}{e}^{4}+60\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{3}{c}^{4}{d}^{5}{e}^{3}+90\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{4}{d}^{6}{e}^{2}+60\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{4}{d}^{7}e-15\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{4}{d}^{8}-118\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+73\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-136\,x{a}^{2}cd{e}^{5}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+36\,xa{c}^{2}{d}^{3}{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+55\,x{c}^{3}{d}^{5}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-48\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{3}{e}^{6}+8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}c{d}^{2}{e}^{4}+10\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{c}^{2}{d}^{4}{e}^{2}+15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(15/2),x)

[Out]

1/192*(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^4*c^4*d^4*e^4+60*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*
d^2)*e)^(1/2))*x^3*c^4*d^5*e^3+90*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^
(1/2))*x^2*c^4*d^6*e^2+60*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x
*c^4*d^7*e-15*x^3*c^3*d^3*e^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+15*arcta
nh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*d^8-118*x^2*a*c^2*d^2*e^4*(c
*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+73*x^2*c^3*d^4*e^2*(c*d*x+a*e)^(1/2)*((a
*e^2-c*d^2)*e)^(1/2)-136*x*a^2*c*d*e^5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)
+36*x*a*c^2*d^3*e^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+55*x*c^3*d^5*e*(c*
d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-48*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1
/2)*a^3*e^6+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d^2*e^4+10*((a*e^2
-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^2*d^4*e^2+15*((a*e^2-c*d^2)*e)^(1/2)*(c*d
*x+a*e)^(1/2)*c^3*d^6)/(e*x+d)^(9/2)/(c*d*x+a*e)^(1/2)/e^3/(a*e^2-c*d^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)^(5/2)/(e*x + d)^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.242277, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(15/2),x, algorithm="fricas")

[Out]

[1/384*(2*(15*c^3*d^3*e^3*x^3 - 15*c^3*d^6 - 10*a*c^2*d^4*e^2 - 8*a^2*c*d^2*e^4
+ 48*a^3*e^6 - (73*c^3*d^4*e^2 - 118*a*c^2*d^2*e^4)*x^2 - (55*c^3*d^5*e + 36*a*c
^2*d^3*e^3 - 136*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr
t(-c*d^2*e + a*e^3)*sqrt(e*x + d) + 15*(c^4*d^4*e^5*x^5 + 5*c^4*d^5*e^4*x^4 + 10
*c^4*d^6*e^3*x^3 + 10*c^4*d^7*e^2*x^2 + 5*c^4*d^8*e*x + c^4*d^9)*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^7*e^3 - a*d^5*e^5 + (c*d^2*e^8 - a*e^10)*x^5 + 5*(c*d^3*e^7 - a*d
*e^9)*x^4 + 10*(c*d^4*e^6 - a*d^2*e^8)*x^3 + 10*(c*d^5*e^5 - a*d^3*e^7)*x^2 + 5*
(c*d^6*e^4 - a*d^4*e^6)*x)*sqrt(-c*d^2*e + a*e^3)), 1/192*((15*c^3*d^3*e^3*x^3 -
15*c^3*d^6 - 10*a*c^2*d^4*e^2 - 8*a^2*c*d^2*e^4 + 48*a^3*e^6 - (73*c^3*d^4*e^2
- 118*a*c^2*d^2*e^4)*x^2 - (55*c^3*d^5*e + 36*a*c^2*d^3*e^3 - 136*a^2*c*d*e^5)*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
) - 15*(c^4*d^4*e^5*x^5 + 5*c^4*d^5*e^4*x^4 + 10*c^4*d^6*e^3*x^3 + 10*c^4*d^7*e^
2*x^2 + 5*c^4*d^8*e*x + c^4*d^9)*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^7*e^3 - a*d^5*e^5 + (c*d^2*e^8 - a*e^10)*x^5 + 5*(c*d^3*e^7 - a*d*
e^9)*x^4 + 10*(c*d^4*e^6 - a*d^2*e^8)*x^3 + 10*(c*d^5*e^5 - a*d^3*e^7)*x^2 + 5*(
c*d^6*e^4 - a*d^4*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)**(5/2)/(e*x+d)**(15/2),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)^(5/2)/(e*x + d)^(15/2),x, algorithm="giac")

[Out]

Timed out