3.2007 \(\int \frac{(d+e x)^{15/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=222 \[ -\frac{63 e^2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{11/2} d^{11/2}}+\frac{63 e^2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{4 c^5 d^5}+\frac{21 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac{9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac{63 e^2 (d+e x)^{5/2}}{20 c^3 d^3} \]

[Out]

(63*e^2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(4*c^5*d^5) + (21*e^2*(c*d^2 - a*e^2)*(
d + e*x)^(3/2))/(4*c^4*d^4) + (63*e^2*(d + e*x)^(5/2))/(20*c^3*d^3) - (9*e*(d +
e*x)^(7/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(9/2)/(2*c*d*(a*e + c*d*x)^2)
- (63*e^2*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d
^2 - a*e^2]])/(4*c^(11/2)*d^(11/2))

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Rubi [A]  time = 0.45219, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{63 e^2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{11/2} d^{11/2}}+\frac{63 e^2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{4 c^5 d^5}+\frac{21 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac{9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac{63 e^2 (d+e x)^{5/2}}{20 c^3 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(15/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(63*e^2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(4*c^5*d^5) + (21*e^2*(c*d^2 - a*e^2)*(
d + e*x)^(3/2))/(4*c^4*d^4) + (63*e^2*(d + e*x)^(5/2))/(20*c^3*d^3) - (9*e*(d +
e*x)^(7/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(9/2)/(2*c*d*(a*e + c*d*x)^2)
- (63*e^2*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d
^2 - a*e^2]])/(4*c^(11/2)*d^(11/2))

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Rubi in Sympy [A]  time = 94.9662, size = 204, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{2 c d \left (a e + c d x\right )^{2}} - \frac{9 e \left (d + e x\right )^{\frac{7}{2}}}{4 c^{2} d^{2} \left (a e + c d x\right )} + \frac{63 e^{2} \left (d + e x\right )^{\frac{5}{2}}}{20 c^{3} d^{3}} - \frac{21 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{4 c^{4} d^{4}} + \frac{63 e^{2} \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}}{4 c^{5} d^{5}} - \frac{63 e^{2} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 c^{\frac{11}{2}} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(15/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

-(d + e*x)**(9/2)/(2*c*d*(a*e + c*d*x)**2) - 9*e*(d + e*x)**(7/2)/(4*c**2*d**2*(
a*e + c*d*x)) + 63*e**2*(d + e*x)**(5/2)/(20*c**3*d**3) - 21*e**2*(d + e*x)**(3/
2)*(a*e**2 - c*d**2)/(4*c**4*d**4) + 63*e**2*sqrt(d + e*x)*(a*e**2 - c*d**2)**2/
(4*c**5*d**5) - 63*e**2*(a*e**2 - c*d**2)**(5/2)*atan(sqrt(c)*sqrt(d)*sqrt(d + e
*x)/sqrt(a*e**2 - c*d**2))/(4*c**(11/2)*d**(11/2))

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Mathematica [A]  time = 0.45345, size = 239, normalized size = 1.08 \[ -\frac{\sqrt{d+e x} \left (-8 e^2 \left (30 a^2 e^4-65 a c d^2 e^2+36 c^2 d^4\right ) (a e+c d x)^2-8 c^2 d^2 e^4 x^2 (a e+c d x)^2+85 e \left (c d^2-a e^2\right )^3 (a e+c d x)+10 \left (c d^2-a e^2\right )^4-8 c d e^3 x \left (7 c d^2-5 a e^2\right ) (a e+c d x)^2\right )}{20 c^5 d^5 (a e+c d x)^2}-\frac{63 e^2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{11/2} d^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(15/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(10*(c*d^2 - a*e^2)^4 + 85*e*(c*d^2 - a*e^2)^3*(a*e + c*d*x) - 8
*e^2*(36*c^2*d^4 - 65*a*c*d^2*e^2 + 30*a^2*e^4)*(a*e + c*d*x)^2 - 8*c*d*e^3*(7*c
*d^2 - 5*a*e^2)*x*(a*e + c*d*x)^2 - 8*c^2*d^2*e^4*x^2*(a*e + c*d*x)^2))/(20*c^5*
d^5*(a*e + c*d*x)^2) - (63*e^2*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sq
rt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*c^(11/2)*d^(11/2))

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Maple [B]  time = 0.027, size = 635, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(15/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

2/5*e^2*(e*x+d)^(5/2)/c^3/d^3-2*e^4/c^4/d^4*(e*x+d)^(3/2)*a+2*e^2/c^3/d^2*(e*x+d
)^(3/2)+12*e^6/c^5/d^5*a^2*(e*x+d)^(1/2)-24*e^4/c^4/d^3*a*(e*x+d)^(1/2)+12*e^2/c
^3/d*(e*x+d)^(1/2)+17/4*e^8/c^4/d^4/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)*a^3-51/4*e^6
/c^3/d^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)*a^2+51/4*e^4/c^2/(c*d*e*x+a*e^2)^2*(e*x
+d)^(3/2)*a-17/4*e^2/c*d^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)+15/4*e^10/c^5/d^5/(c*
d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a^4-15*e^8/c^4/d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*
a^3+45/2*e^6/c^3/d/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a^2-15*e^4/c^2*d/(c*d*e*x+a*e
^2)^2*(e*x+d)^(1/2)*a+15/4*e^2/c*d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)-63/4*e^8/c^
5/d^5/((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+189/4*e^6/c^4/d^3/((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^2-189/4*e^4/c^3/d/((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+63/4*e^2/c^2*d/((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)^(15/2)/(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 = 0.238697, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(15/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

[1/40*(315*(a^2*c^2*d^4*e^4 - 2*a^3*c*d^2*e^6 + a^4*e^8 + (c^4*d^6*e^2 - 2*a*c^3
*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 - 2*a^2*c^2*d^3*e^5 + a^3*c*d
*e^7)*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^4*d^4*e^4*x^4 - 1
0*c^4*d^8 - 45*a*c^3*d^6*e^2 + 483*a^2*c^2*d^4*e^4 - 735*a^3*c*d^2*e^6 + 315*a^4
*e^8 + 8*(7*c^4*d^5*e^3 - 3*a*c^3*d^3*e^5)*x^3 + 24*(12*c^4*d^6*e^2 - 17*a*c^3*d
^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 - (85*c^4*d^7*e - 831*a*c^3*d^5*e^3 + 1239*a^2*c
^2*d^3*e^5 - 525*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x +
a^2*c^5*d^5*e^2), -1/20*(315*(a^2*c^2*d^4*e^4 - 2*a^3*c*d^2*e^6 + a^4*e^8 + (c^
4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 - 2*a^2*c^
2*d^3*e^5 + a^3*c*d*e^7)*x)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e*x + d)/sq
rt(-(c*d^2 - a*e^2)/(c*d))) - (8*c^4*d^4*e^4*x^4 - 10*c^4*d^8 - 45*a*c^3*d^6*e^2
+ 483*a^2*c^2*d^4*e^4 - 735*a^3*c*d^2*e^6 + 315*a^4*e^8 + 8*(7*c^4*d^5*e^3 - 3*
a*c^3*d^3*e^5)*x^3 + 24*(12*c^4*d^6*e^2 - 17*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*
x^2 - (85*c^4*d^7*e - 831*a*c^3*d^5*e^3 + 1239*a^2*c^2*d^3*e^5 - 525*a^3*c*d*e^7
)*x)*sqrt(e*x + d))/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*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)**(15/2)/(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)^(15/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Timed out