3.2008 \(\int \frac{(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=186 \[ -\frac{35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}+\frac{35 e^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3} \]

[Out]

(35*e^2*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(4*c^4*d^4) + (35*e^2*(d + e*x)^(3/2))/(1
2*c^3*d^3) - (7*e*(d + e*x)^(5/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(7/2)/(
2*c*d*(a*e + c*d*x)^2) - (35*e^2*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*
Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*c^(9/2)*d^(9/2))

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Rubi [A]  time = 0.365569, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}+\frac{35 e^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(35*e^2*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(4*c^4*d^4) + (35*e^2*(d + e*x)^(3/2))/(1
2*c^3*d^3) - (7*e*(d + e*x)^(5/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(7/2)/(
2*c*d*(a*e + c*d*x)^2) - (35*e^2*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*
Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*c^(9/2)*d^(9/2))

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Rubi in Sympy [A]  time = 76.8625, size = 170, normalized size = 0.91 \[ - \frac{\left (d + e x\right )^{\frac{7}{2}}}{2 c d \left (a e + c d x\right )^{2}} - \frac{7 e \left (d + e x\right )^{\frac{5}{2}}}{4 c^{2} d^{2} \left (a e + c d x\right )} + \frac{35 e^{2} \left (d + e x\right )^{\frac{3}{2}}}{12 c^{3} d^{3}} - \frac{35 e^{2} \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{4 c^{4} d^{4}} + \frac{35 e^{2} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 c^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

-(d + e*x)**(7/2)/(2*c*d*(a*e + c*d*x)**2) - 7*e*(d + e*x)**(5/2)/(4*c**2*d**2*(
a*e + c*d*x)) + 35*e**2*(d + e*x)**(3/2)/(12*c**3*d**3) - 35*e**2*sqrt(d + e*x)*
(a*e**2 - c*d**2)/(4*c**4*d**4) + 35*e**2*(a*e**2 - c*d**2)**(3/2)*atan(sqrt(c)*
sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))/(4*c**(9/2)*d**(9/2))

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Mathematica [A]  time = 0.394178, size = 188, normalized size = 1.01 \[ -\frac{35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}-\frac{\sqrt{d+e x} \left (8 e^2 \left (9 a e^2-10 c d^2\right ) (a e+c d x)^2+39 e \left (c d^2-a e^2\right )^2 (a e+c d x)+6 \left (c d^2-a e^2\right )^3-8 c d e^3 x (a e+c d x)^2\right )}{12 c^4 d^4 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(6*(c*d^2 - a*e^2)^3 + 39*e*(c*d^2 - a*e^2)^2*(a*e + c*d*x) + 8*
e^2*(-10*c*d^2 + 9*a*e^2)*(a*e + c*d*x)^2 - 8*c*d*e^3*x*(a*e + c*d*x)^2))/(12*c^
4*d^4*(a*e + c*d*x)^2) - (35*e^2*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*
Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*c^(9/2)*d^(9/2))

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Maple [B]  time = 0.025, size = 449, normalized size = 2.4 \[{\frac{2\,{e}^{2}}{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-6\,{\frac{{e}^{4}a\sqrt{ex+d}}{{c}^{4}{d}^{4}}}+6\,{\frac{{e}^{2}\sqrt{ex+d}}{{c}^{3}{d}^{2}}}-{\frac{13\,{e}^{6}{a}^{2}}{4\,{c}^{3}{d}^{3} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{e}^{4}a}{2\,{c}^{2}d \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{13\,d{e}^{2}}{4\,c \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{11\,{e}^{8}{a}^{3}}{4\,{c}^{4}{d}^{4} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{33\,{e}^{6}{a}^{2}}{4\,{c}^{3}{d}^{2} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{33\,{e}^{4}a}{4\,{c}^{2} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{11\,{d}^{2}{e}^{2}}{4\,c \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{e}^{6}{a}^{2}}{4\,{c}^{4}{d}^{4}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}}-{\frac{35\,{e}^{4}a}{2\,{c}^{3}{d}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}}+{\frac{35\,{e}^{2}}{4\,{c}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(13/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

2/3*e^2*(e*x+d)^(3/2)/c^3/d^3-6*e^4/c^4/d^4*a*(e*x+d)^(1/2)+6*e^2/c^3/d^2*(e*x+d
)^(1/2)-13/4*e^6/c^3/d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)*a^2+13/2*e^4/c^2/d/(c*d
*e*x+a*e^2)^2*(e*x+d)^(3/2)*a-13/4*e^2/c*d/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)-11/4*
e^8/c^4/d^4/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a^3+33/4*e^6/c^3/d^2/(c*d*e*x+a*e^2)
^2*(e*x+d)^(1/2)*a^2-33/4*e^4/c^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a+11/4*e^2/c*d
^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)+35/4*e^6/c^4/d^4/((a*e^2-c*d^2)*c*d)^(1/2)*ar
ctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2))*a^2-35/2*e^4/c^3/d^2/((a*e^2-c
*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2))*a+35/4*e^2/
c^2/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(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((e*x + d)^(13/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227485, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \,{\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} -{\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac{105 \,{\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \,{\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} -{\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{12 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(13/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

[1/24*(105*(a^2*c*d^2*e^4 - a^3*e^6 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 2*(a*c
^2*d^3*e^3 - a^2*c*d*e^5)*x)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2
- a*e^2 - 2*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*(8
*c^3*d^3*e^3*x^3 - 6*c^3*d^6 - 21*a*c^2*d^4*e^2 + 140*a^2*c*d^2*e^4 - 105*a^3*e^
6 + 8*(10*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - (39*c^3*d^5*e - 238*a*c^2*d^3*e^3
+ 175*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d
^4*e^2), -1/12*(105*(a^2*c*d^2*e^4 - a^3*e^6 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2
+ 2*(a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e
*x + d)/sqrt(-(c*d^2 - a*e^2)/(c*d))) - (8*c^3*d^3*e^3*x^3 - 6*c^3*d^6 - 21*a*c^
2*d^4*e^2 + 140*a^2*c*d^2*e^4 - 105*a^3*e^6 + 8*(10*c^3*d^4*e^2 - 7*a*c^2*d^2*e^
4)*x^2 - (39*c^3*d^5*e - 238*a*c^2*d^3*e^3 + 175*a^2*c*d*e^5)*x)*sqrt(e*x + d))/
(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2)]

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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((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,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((e*x + d)^(13/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Timed out