3.2034 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=315 \[ \frac{3 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}}+\frac{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]

[Out]

-(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e^2*(d + e*x)^(7/2)) + (c^
2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*e^2*(c*d^2 - a*e^2)*(d +
e*x)^(5/2)) + (3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^2*(c
*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)
/(4*e*(d + e*x)^(11/2)) + (3*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^(5/2)*(c*d^2 - a*e
^2)^(5/2))

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Rubi [A]  time = 0.651814, antiderivative size = 315, 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 \[ \frac{3 c^4 d^4 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}}+\frac{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]

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)^(13/2),x]

[Out]

-(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e^2*(d + e*x)^(7/2)) + (c^
2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*e^2*(c*d^2 - a*e^2)*(d +
e*x)^(5/2)) + (3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^2*(c
*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)
/(4*e*(d + e*x)^(11/2)) + (3*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^(5/2)*(c*d^2 - a*e
^2)^(5/2))

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Rubi in Sympy [A]  time = 135.058, size = 294, normalized size = 0.93 \[ - \frac{3 c^{4} d^{4} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{64 e^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{3 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{32 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 e^{2} \left (d + e x\right )^{\frac{7}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 e \left (d + e x\right )^{\frac{11}{2}}} \]

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)**(13/2),x)

[Out]

-3*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**(5/2)*(a*e**2 - c*d**2)**(5/2)) + 3*c**3
*d**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(64*e**2*(d + e*x)**(3/2)*(
a*e**2 - c*d**2)**2) - c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/
(32*e**2*(d + e*x)**(5/2)*(a*e**2 - c*d**2)) - c*d*sqrt(a*d*e + c*d*e*x**2 + x*(
a*e**2 + c*d**2))/(8*e**2*(d + e*x)**(7/2)) - (a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))**(3/2)/(4*e*(d + e*x)**(11/2))

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Mathematica [A]  time = 0.83102, size = 231, normalized size = 0.73 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{-16 a^3 e^6+24 a^2 c d e^4 (d-e x)-2 a c^2 d^2 e^2 \left (d^2-22 d e x+e^2 x^2\right )+c^3 d^3 \left (-3 d^3-11 d^2 e x+11 d e^2 x^2+3 e^3 x^3\right )}{(d+e x)^4 \left (c d^2 e-a e^3\right )^2 (a e+c d x)}-\frac{3 c^4 d^4 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{e^{5/2} \left (a e^2-c d^2\right )^{5/2} (a e+c d x)^{3/2}}\right )}{64 (d+e x)^{3/2}} \]

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)^(13/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((-16*a^3*e^6 + 24*a^2*c*d*e^4*(d - e*x) - 2*a*
c^2*d^2*e^2*(d^2 - 22*d*e*x + e^2*x^2) + c^3*d^3*(-3*d^3 - 11*d^2*e*x + 11*d*e^2
*x^2 + 3*e^3*x^3))/((c*d^2*e - a*e^3)^2*(a*e + c*d*x)*(d + e*x)^4) - (3*c^4*d^4*
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)^(5/2)*(a*e + c*d*x)^(3/2))))/(64*(d + e*x)^(3/2))

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Maple [B]  time = 0.046, size = 662, normalized size = 2.1 \[ -{\frac{1}{64\,{e}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{4}{c}^{4}{d}^{4}{e}^{4}+12\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{3}{c}^{4}{d}^{5}{e}^{3}+18\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{4}{d}^{6}{e}^{2}+12\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{4}{d}^{7}e-3\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{4}{d}^{8}+2\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-11\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+24\,x{a}^{2}cd{e}^{5}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-44\,xa{c}^{2}{d}^{3}{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+11\,x{c}^{3}{d}^{5}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+16\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{3}{e}^{6}-24\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}c{d}^{2}{e}^{4}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{c}^{2}{d}^{4}{e}^{2}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}{\frac{1}{\sqrt{cdx+ae}}}} \]

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)^(13/2),x)

[Out]

-1/64*(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^4*c^4*d^4*e^4+12*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+18*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+12*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*
c^4*d^7*e-3*x^3*c^3*d^3*e^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+3*arctanh(
e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*d^8+2*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)-11*x^2*c^3*d^4*e^2*(c*d*x+a*e)^(1/2)*((a*e^2-
c*d^2)*e)^(1/2)+24*x*a^2*c*d*e^5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-44*x*
a*c^2*d^3*e^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+11*x*c^3*d^5*e*(c*d*x+a*
e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+16*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^
3*e^6-24*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d^2*e^4+2*((a*e^2-c*d^2
)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^2*d^4*e^2+3*((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)/((a*e^2-c*d^2)*e)^(1/2)/e^2/(a*e^2-c*d^2)^2/(c*d*x
+a*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)^(3/2)/(e*x + d)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240919, 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)^(3/2)/(e*x + d)^(13/2),x, algorithm="fricas")

[Out]

[1/128*(2*(3*c^3*d^3*e^3*x^3 - 3*c^3*d^6 - 2*a*c^2*d^4*e^2 + 24*a^2*c*d^2*e^4 -
16*a^3*e^6 + (11*c^3*d^4*e^2 - 2*a*c^2*d^2*e^4)*x^2 - (11*c^3*d^5*e - 44*a*c^2*d
^3*e^3 + 24*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) + 3*(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^2*d^9*e^2 - 2*a*c*d^7*e^4 + a^2*d^5*e^6 + (c^2*d^4*e^7 - 2*a*c*d^2*e^9 +
a^2*e^11)*x^5 + 5*(c^2*d^5*e^6 - 2*a*c*d^3*e^8 + a^2*d*e^10)*x^4 + 10*(c^2*d^6*e
^5 - 2*a*c*d^4*e^7 + a^2*d^2*e^9)*x^3 + 10*(c^2*d^7*e^4 - 2*a*c*d^5*e^6 + a^2*d^
3*e^8)*x^2 + 5*(c^2*d^8*e^3 - 2*a*c*d^6*e^5 + a^2*d^4*e^7)*x)*sqrt(-c*d^2*e + a*
e^3)), 1/64*((3*c^3*d^3*e^3*x^3 - 3*c^3*d^6 - 2*a*c^2*d^4*e^2 + 24*a^2*c*d^2*e^4
- 16*a^3*e^6 + (11*c^3*d^4*e^2 - 2*a*c^2*d^2*e^4)*x^2 - (11*c^3*d^5*e - 44*a*c^
2*d^3*e^3 + 24*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) - 3*(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^2*d^9*e^2 - 2*a*c*d^7*e^4 + a^2*d^5*e^6
+ (c^2*d^4*e^7 - 2*a*c*d^2*e^9 + a^2*e^11)*x^5 + 5*(c^2*d^5*e^6 - 2*a*c*d^3*e^8
+ a^2*d*e^10)*x^4 + 10*(c^2*d^6*e^5 - 2*a*c*d^4*e^7 + a^2*d^2*e^9)*x^3 + 10*(c^2
*d^7*e^4 - 2*a*c*d^5*e^6 + a^2*d^3*e^8)*x^2 + 5*(c^2*d^8*e^3 - 2*a*c*d^6*e^5 + a
^2*d^4*e^7)*x)*sqrt(c*d^2*e - a*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)**(3/2)/(e*x+d)**(13/2),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)^(3/2)/(e*x + d)^(13/2),x, algorithm="giac")

[Out]

Timed out