3.2340 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=225 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-3*(b^2 - 4*a*c)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*
d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x
+ c*x^2)^(3/2))/(8*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + (3*(b^2 - 4*a*c)^2*Arc
Tanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x
+ c*x^2])])/(128*(c*d^2 - b*d*e + a*e^2)^(5/2))

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Rubi [A]  time = 0.489402, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

(-3*(b^2 - 4*a*c)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*
d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x
+ c*x^2)^(3/2))/(8*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + (3*(b^2 - 4*a*c)^2*Arc
Tanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x
+ c*x^2])])/(128*(c*d^2 - b*d*e + a*e^2)^(5/2))

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Rubi in Sympy [A]  time = 61.9289, size = 209, normalized size = 0.93 \[ - \frac{3 \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{128 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{64 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{8 \left (d + e x\right )^{4} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**5,x)

[Out]

-3*(-4*a*c + b**2)**2*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b*x + c*
x**2)*sqrt(a*e**2 - b*d*e + c*d**2)))/(128*(a*e**2 - b*d*e + c*d**2)**(5/2)) + 3
*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)*(2*a*e - b*d + x*(b*e - 2*c*d))/(64*(d +
e*x)**2*(a*e**2 - b*d*e + c*d**2)**2) - (a + b*x + c*x**2)**(3/2)*(2*a*e - b*d
+ x*(b*e - 2*c*d))/(8*(d + e*x)**4*(a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 1.17253, size = 331, normalized size = 1.47 \[ \frac{-2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (2 (d+e x)^2 \left (4 c e (5 a e-6 b d)+b^2 e^2+24 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )-(d+e x)^3 (2 c d-b e) \left (4 c e (5 a e-2 b d)-3 b^2 e^2+8 c^2 d^2\right )-24 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2+16 \left (e (a e-b d)+c d^2\right )^3\right )-3 e^3 \left (b^2-4 a c\right )^2 (d+e x)^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+3 e^3 \left (b^2-4 a c\right )^2 (d+e x)^4 \log (d+e x)}{128 e^3 (d+e x)^4 \left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

(-2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(16*(c*d^2 + e*(-(b*d)
+ a*e))^3 - 24*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x) + 2*(c*d^2 +
e*(-(b*d) + a*e))*(24*c^2*d^2 + b^2*e^2 + 4*c*e*(-6*b*d + 5*a*e))*(d + e*x)^2 -
(2*c*d - b*e)*(8*c^2*d^2 - 3*b^2*e^2 + 4*c*e*(-2*b*d + 5*a*e))*(d + e*x)^3) + 3
*(b^2 - 4*a*c)^2*e^3*(d + e*x)^4*Log[d + e*x] - 3*(b^2 - 4*a*c)^2*e^3*(d + e*x)^
4*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a
+ x*(b + c*x)]])/(128*e^3*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*(d + e*x)^4)

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Maple [B]  time = 0.033, size = 15932, normalized size = 70.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x)

[Out]

result too large to display

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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*x^2 + b*x + a)^(3/2)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.96749, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[1/256*(4*(24*a^2*b*d*e^2 - 16*a^3*e^3 - (3*b^3 - 20*a*b*c)*d^3 - 2*(a*b^2 + 20*
a^2*c)*d^2*e + (16*c^3*d^3 - 24*b*c^2*d^2*e + 2*(b^2*c + 20*a*c^2)*d*e^2 + (3*b^
3 - 20*a*b*c)*e^3)*x^3 + (24*b*c^2*d^3 - 4*(11*b^2*c - 8*a*c^2)*d^2*e + (11*b^3
+ 28*a*b*c)*d*e^2 - 2*(a*b^2 + 20*a^2*c)*e^3)*x^2 - (24*a^2*b*e^3 - 2*(b^2*c + 2
0*a*c^2)*d^3 + (11*b^3 + 28*a*b*c)*d^2*e - 4*(11*a*b^2 - 8*a^2*c)*d*e^2)*x)*sqrt
(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) + 3*((b^4 - 8*a*b^2*c + 16*a^2*c^2
)*e^4*x^4 + 4*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d*e^3*x^3 + 6*(b^4 - 8*a*b^2*c + 16
*a^2*c^2)*d^2*e^2*x^2 + 4*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3*e*x + (b^4 - 8*a*b^
2*c + 16*a^2*c^2)*d^4)*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*
d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4
*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) - 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3
- (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^
2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d^8 - 2*b*c*d^7*e
- 2*a*b*d^5*e^3 + a^2*d^4*e^4 + (b^2 + 2*a*c)*d^6*e^2 + (c^2*d^4*e^4 - 2*b*c*d^
3*e^5 - 2*a*b*d*e^7 + a^2*e^8 + (b^2 + 2*a*c)*d^2*e^6)*x^4 + 4*(c^2*d^5*e^3 - 2*
b*c*d^4*e^4 - 2*a*b*d^2*e^6 + a^2*d*e^7 + (b^2 + 2*a*c)*d^3*e^5)*x^3 + 6*(c^2*d^
6*e^2 - 2*b*c*d^5*e^3 - 2*a*b*d^3*e^5 + a^2*d^2*e^6 + (b^2 + 2*a*c)*d^4*e^4)*x^2
+ 4*(c^2*d^7*e - 2*b*c*d^6*e^2 - 2*a*b*d^4*e^4 + a^2*d^3*e^5 + (b^2 + 2*a*c)*d^
5*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/128*(2*(24*a^2*b*d*e^2 - 16*a^3*e^3 -
(3*b^3 - 20*a*b*c)*d^3 - 2*(a*b^2 + 20*a^2*c)*d^2*e + (16*c^3*d^3 - 24*b*c^2*d^2
*e + 2*(b^2*c + 20*a*c^2)*d*e^2 + (3*b^3 - 20*a*b*c)*e^3)*x^3 + (24*b*c^2*d^3 -
4*(11*b^2*c - 8*a*c^2)*d^2*e + (11*b^3 + 28*a*b*c)*d*e^2 - 2*(a*b^2 + 20*a^2*c)*
e^3)*x^2 - (24*a^2*b*e^3 - 2*(b^2*c + 20*a*c^2)*d^3 + (11*b^3 + 28*a*b*c)*d^2*e
- 4*(11*a*b^2 - 8*a^2*c)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x
+ a) - 3*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4*x^4 + 4*(b^4 - 8*a*b^2*c + 16*a^2*
c^2)*d*e^3*x^3 + 6*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2*e^2*x^2 + 4*(b^4 - 8*a*b^2
*c + 16*a^2*c^2)*d^3*e*x + (b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^4)*arctan(-1/2*sqrt(
-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)
*sqrt(c*x^2 + b*x + a))))/((c^2*d^8 - 2*b*c*d^7*e - 2*a*b*d^5*e^3 + a^2*d^4*e^4
+ (b^2 + 2*a*c)*d^6*e^2 + (c^2*d^4*e^4 - 2*b*c*d^3*e^5 - 2*a*b*d*e^7 + a^2*e^8 +
(b^2 + 2*a*c)*d^2*e^6)*x^4 + 4*(c^2*d^5*e^3 - 2*b*c*d^4*e^4 - 2*a*b*d^2*e^6 + a
^2*d*e^7 + (b^2 + 2*a*c)*d^3*e^5)*x^3 + 6*(c^2*d^6*e^2 - 2*b*c*d^5*e^3 - 2*a*b*d
^3*e^5 + a^2*d^2*e^6 + (b^2 + 2*a*c)*d^4*e^4)*x^2 + 4*(c^2*d^7*e - 2*b*c*d^6*e^2
- 2*a*b*d^4*e^4 + a^2*d^3*e^5 + (b^2 + 2*a*c)*d^5*e^3)*x)*sqrt(-c*d^2 + b*d*e -
a*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((c*x**2+b*x+a)**(3/2)/(e*x+d)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 11.5552, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^5,x, algorithm="giac")

[Out]

Done