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

Optimal. Leaf size=886 \[ \frac{\tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^{5/2}}{e^5 (e f-d g)}+\frac{\left (c x^2+b x+a\right )^{3/2} \left (c d^2-b e d+a e^2\right )}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a} \left (c d^2-b e d+a e^2\right )}{8 c e^4 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left (c f^2-b g f+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{g^5 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)} \]

[Out]

((c*d^2 - b*d*e + a*e^2)*(8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2
*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*e^4*(e*f - d*g)) - ((64*c^3*e*f^4 - 1
6*c^2*e*f^2*g*(9*b*f - 8*a*g) - b^2*g^3*(5*b*e*f + 3*b*d*g - 8*a*e*g) + 4*c*g^2*
(22*b^2*e*f^2 + 16*a^2*e*g^2 - 3*a*b*g*(13*e*f - d*g)) - 2*c*g*(16*c^2*e*f^3 + b
*g^2*(5*b*e*f + 3*b*d*g - 8*a*e*g) - 4*c*g*(6*b*e*f^2 - a*g*(7*e*f - 3*d*g)))*x)
*Sqrt[a + b*x + c*x^2])/(64*c*e*g^4*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)*(a +
b*x + c*x^2)^(3/2))/(3*e^2*(e*f - d*g)) - ((8*c*e*f^2 - g*(11*b*e*f - 3*b*d*g -
8*a*e*g) - 6*c*g*(e*f - d*g)*x)*(a + b*x + c*x^2)^(3/2))/(24*e*g^2*(e*f - d*g))
- ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d -
3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*e^5*
(e*f - d*g)) + ((128*c^4*e*f^5 - 320*c^3*e*f^3*g*(b*f - a*g) - b^3*g^4*(5*b*e*f
+ 3*b*d*g - 8*a*e*g) + 48*c^2*g^2*(5*b^2*e*f^3 - 10*a*b*e*f^2*g + a^2*g^2*(5*e*f
- d*g)) - 8*b*c*g^3*(5*b^2*e*f^2 + 12*a^2*e*g^2 - 3*a*b*g*(5*e*f + d*g)))*ArcTa
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3/2)*e*g^5*(e*f - d*g
)) + ((c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*S
qrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e^5*(e*f - d*g)) - ((c*f^2
- b*f*g + a*g^2)^(5/2)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b
*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(g^5*(e*f - d*g))

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Rubi [A]  time = 3.88809, antiderivative size = 886, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ \frac{\tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^{5/2}}{e^5 (e f-d g)}+\frac{\left (c x^2+b x+a\right )^{3/2} \left (c d^2-b e d+a e^2\right )}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a} \left (c d^2-b e d+a e^2\right )}{8 c e^4 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left (c f^2-b g f+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{g^5 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)} \]

Antiderivative was successfully verified.

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

[Out]

((c*d^2 - b*d*e + a*e^2)*(8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2
*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*e^4*(e*f - d*g)) - ((64*c^3*e*f^4 - 1
6*c^2*e*f^2*g*(9*b*f - 8*a*g) - b^2*g^3*(5*b*e*f + 3*b*d*g - 8*a*e*g) + 4*c*g^2*
(22*b^2*e*f^2 + 16*a^2*e*g^2 - 3*a*b*g*(13*e*f - d*g)) - 2*c*g*(16*c^2*e*f^3 + b
*g^2*(5*b*e*f + 3*b*d*g - 8*a*e*g) - 4*c*g*(6*b*e*f^2 - a*g*(7*e*f - 3*d*g)))*x)
*Sqrt[a + b*x + c*x^2])/(64*c*e*g^4*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)*(a +
b*x + c*x^2)^(3/2))/(3*e^2*(e*f - d*g)) - ((8*c*e*f^2 - g*(11*b*e*f - 3*b*d*g -
8*a*e*g) - 6*c*g*(e*f - d*g)*x)*(a + b*x + c*x^2)^(3/2))/(24*e*g^2*(e*f - d*g))
- ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d -
3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*e^5*
(e*f - d*g)) + ((128*c^4*e*f^5 - 320*c^3*e*f^3*g*(b*f - a*g) - b^3*g^4*(5*b*e*f
+ 3*b*d*g - 8*a*e*g) + 48*c^2*g^2*(5*b^2*e*f^3 - 10*a*b*e*f^2*g + a^2*g^2*(5*e*f
- d*g)) - 8*b*c*g^3*(5*b^2*e*f^2 + 12*a^2*e*g^2 - 3*a*b*g*(5*e*f + d*g)))*ArcTa
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3/2)*e*g^5*(e*f - d*g
)) + ((c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*S
qrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e^5*(e*f - d*g)) - ((c*f^2
- b*f*g + a*g^2)^(5/2)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b
*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(g^5*(e*f - d*g))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 3.08084, size = 749, normalized size = 0.85 \[ \frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (240 c^2 e^2 g^2 \left (a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )-40 b^2 c e^3 g^3 (-3 a e g+b d g+b e f)-320 c^3 e g \left (b \left (d^3 g^3+d^2 e f g^2+d e^2 f^2 g+e^3 f^3\right )-a e g \left (d^2 g^2+d e f g+e^2 f^2\right )\right )-5 b^4 e^4 g^4+128 c^4 \left (d^4 g^4+d^3 e f g^3+d^2 e^2 f^2 g^2+d e^3 f^3 g+e^4 f^4\right )\right )}{128 c^{3/2} e^5 g^5}+\frac{\sqrt{a+x (b+c x)} \left (8 c^2 e g \left (a e g (-56 d g-56 e f+27 e g x)+b \left (54 d^2 g^2+2 d e g (27 f-13 g x)+e^2 \left (54 f^2-26 f g x+17 g^2 x^2\right )\right )\right )+2 b c e^2 g^2 (278 a e g+b (-132 d g-132 e f+59 e g x))+15 b^3 e^3 g^3-16 c^3 \left (12 d^3 g^3-6 d^2 e g^2 (g x-2 f)+2 d e^2 g \left (6 f^2-3 f g x+2 g^2 x^2\right )+e^3 \left (12 f^3-6 f^2 g x+4 f g^2 x^2-3 g^3 x^3\right )\right )\right )}{192 c e^4 g^4}+\frac{\log (d+e x) \left (e (a e-b d)+c d^2\right )^{5/2}}{e^5 (e f-d g)}-\frac{\left (e (a e-b d)+c d^2\right )^{5/2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{e^5 (e f-d g)}+\frac{\log (f+g x) \left (g (a g-b f)+c f^2\right )^{5/2}}{g^5 (d g-e f)}+\frac{\left (g (a g-b f)+c f^2\right )^{5/2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}+2 a g-b f+b g x-2 c f x\right )}{g^5 (e f-d g)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(15*b^3*e^3*g^3 + 2*b*c*e^2*g^2*(278*a*e*g + b*(-132*e*f
- 132*d*g + 59*e*g*x)) - 16*c^3*(12*d^3*g^3 - 6*d^2*e*g^2*(-2*f + g*x) + 2*d*e^2
*g*(6*f^2 - 3*f*g*x + 2*g^2*x^2) + e^3*(12*f^3 - 6*f^2*g*x + 4*f*g^2*x^2 - 3*g^3
*x^3)) + 8*c^2*e*g*(a*e*g*(-56*e*f - 56*d*g + 27*e*g*x) + b*(54*d^2*g^2 + 2*d*e*
g*(27*f - 13*g*x) + e^2*(54*f^2 - 26*f*g*x + 17*g^2*x^2)))))/(192*c*e^4*g^4) + (
(c*d^2 + e*(-(b*d) + a*e))^(5/2)*Log[d + e*x])/(e^5*(e*f - d*g)) + ((c*f^2 + g*(
-(b*f) + a*g))^(5/2)*Log[f + g*x])/(g^5*(-(e*f) + d*g)) + ((-5*b^4*e^4*g^4 - 40*
b^2*c*e^3*g^3*(b*e*f + b*d*g - 3*a*e*g) + 128*c^4*(e^4*f^4 + d*e^3*f^3*g + d^2*e
^2*f^2*g^2 + d^3*e*f*g^3 + d^4*g^4) + 240*c^2*e^2*g^2*(a^2*e^2*g^2 - 2*a*b*e*g*(
e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) - 320*c^3*e*g*(-(a*e*g*(e^2*f^2
+ d*e*f*g + d^2*g^2)) + b*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)))*Log[
b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(128*c^(3/2)*e^5*g^5) - ((c*d^2 +
e*(-(b*d) + a*e))^(5/2)*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)]])/(e^5*(e*f - d*g)) + ((c*f^2 + g*(-(b*f)
+ a*g))^(5/2)*Log[-(b*f) + 2*a*g - 2*c*f*x + b*g*x + 2*Sqrt[c*f^2 + g*(-(b*f) +
a*g)]*Sqrt[a + x*(b + c*x)]])/(g^5*(e*f - d*g))

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Maple [B]  time = 0.034, size = 9052, normalized size = 10.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(e*x+d)/(g*x+f),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)^(5/2)/((e*x + d)*(g*x + f)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^(5/2)/((e*x + d)*(g*x + f)),x, algorithm="fricas")

[Out]

Timed out

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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)**(5/2)/(e*x+d)/(g*x+f),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError