3.1989 \(\int \frac{(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=180 \[ -\frac{2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 c^3 d^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d} \]

[Out]

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

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Rubi [A]  time = 0.536876, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 c^3 d^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 77.2622, size = 162, normalized size = 0.9 \[ \frac{2 \left (d + e x\right )^{\frac{7}{2}}}{7 c d} - \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )}{5 c^{2} d^{2}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{3 c^{3} d^{3}} - \frac{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3}}{c^{4} d^{4}} + \frac{2 \left (a e^{2} - c d^{2}\right )^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.240665, size = 179, normalized size = 0.99 \[ \frac{2 \sqrt{d+e x} \left (-105 a^3 e^6+35 a^2 c d e^4 (10 d+e x)-7 a c^2 d^2 e^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )+c^3 d^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 c^4 d^4}-\frac{2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*Sqrt[d + e*x]*(-105*a^3*e^6 + 35*a^2*c*d*e^4*(10*d + e*x) - 7*a*c^2*d^2*e^2*(
58*d^2 + 16*d*e*x + 3*e^2*x^2) + c^3*d^3*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 +
15*e^3*x^3)))/(105*c^4*d^4) - (2*(c*d^2 - a*e^2)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]
*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(9/2)*d^(9/2))

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Maple [B]  time = 0.022, size = 455, normalized size = 2.5 \[{\frac{2}{7\,cd} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{2\,a{e}^{2}}{5\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2}{5\,c} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}{e}^{4}}{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{4\,a{e}^{2}}{3\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,d}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}{e}^{6}\sqrt{ex+d}}{{c}^{4}{d}^{4}}}+6\,{\frac{{a}^{2}{e}^{4}\sqrt{ex+d}}{{c}^{3}{d}^{2}}}-6\,{\frac{a{e}^{2}\sqrt{ex+d}}{{c}^{2}}}+2\,{\frac{{d}^{2}\sqrt{ex+d}}{c}}+2\,{\frac{{a}^{4}{e}^{8}}{{c}^{4}{d}^{4}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-8\,{\frac{{a}^{3}{e}^{6}}{{c}^{3}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+12\,{\frac{{a}^{2}{e}^{4}}{{c}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-8\,{\frac{a{d}^{2}{e}^{2}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+2\,{\frac{{d}^{4}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

2/7*(e*x+d)^(7/2)/c/d-2/5/c^2/d^2*(e*x+d)^(5/2)*a*e^2+2/5/c*(e*x+d)^(5/2)+2/3/c^
3/d^3*(e*x+d)^(3/2)*a^2*e^4-4/3/c^2/d*(e*x+d)^(3/2)*a*e^2+2/3/c*d*(e*x+d)^(3/2)-
2/c^4/d^4*a^3*e^6*(e*x+d)^(1/2)+6/c^3/d^2*a^2*e^4*(e*x+d)^(1/2)-6/c^2*a*e^2*(e*x
+d)^(1/2)+2/c*d^2*(e*x+d)^(1/2)+2/c^4/d^4/((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^4*e^8-8/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^3*e^6+12/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))*a^2*e^4-8/c
*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*e^2+2*d^4/((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)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257927, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \,{\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{105 \, c^{4} d^{4}}, -\frac{2 \,{\left (105 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \,{\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, c^{4} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

[1/105*(105*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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*(15*c^3*d^3*e^3*x^3 + 176*c^3*d^6 - 406*a*c
^2*d^4*e^2 + 350*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(22*c^3*d^4*e^2 - 7*a*c^2*d^2*e
^4)*x^2 + (122*c^3*d^5*e - 112*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(e*x + d))
/(c^4*d^4), -2/105*(105*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*
sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e*x + d)/sqrt(-(c*d^2 - a*e^2)/(c*d)))
- (15*c^3*d^3*e^3*x^3 + 176*c^3*d^6 - 406*a*c^2*d^4*e^2 + 350*a^2*c*d^2*e^4 - 10
5*a^3*e^6 + 3*(22*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (122*c^3*d^5*e - 112*a*c^
2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^4*d^4)]

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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)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**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((e*x + d)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

Timed out