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

Optimal. Leaf size=662 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (96 c^3 e^2 \left (-a^2 e^2 g (2 e f-d g)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right )+16 b c^2 e^3 \left (3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right )-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right )}{256 c^{7/2} e^6}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (3 b^2 e^2 g^2-6 c e g x (-b e g-2 c d g+4 c e f)-6 b c e g (2 e f-d g)-16 c^2 (e f-d g)^2\right )}{48 c^2 e^3}+\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (g \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )-6 b^2 c e^3 g (2 a e g-b d g+2 b e f)-32 c^3 e (5 b d-4 a e) (e f-d g)^2+8 b c^2 e^2 \left (3 a e g (2 e f-d g)+2 b (e f-d g)^2\right )+3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2\right )}{128 c^3 e^5}+\frac{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^6}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e} \]

[Out]

((3*b^4*e^4*g^2 + 128*c^4*d^2*(e*f - d*g)^2 - 32*c^3*e*(5*b*d - 4*a*e)*(e*f - d*
g)^2 - 6*b^2*c*e^3*g*(2*b*e*f - b*d*g + 2*a*e*g) + 8*b*c^2*e^2*(2*b*(e*f - d*g)^
2 + 3*a*e*g*(2*e*f - d*g)) + 2*c*e*((16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a
*e))*g*(4*c*e*f - 2*c*d*g - b*e*g) - 8*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*
x)*Sqrt[a + b*x + c*x^2])/(128*c^3*e^5) - ((3*b^2*e^2*g^2 - 16*c^2*(e*f - d*g)^2
- 6*b*c*e*g*(2*e*f - d*g) - 6*c*e*g*(4*c*e*f - 2*c*d*g - b*e*g)*x)*(a + b*x + c
*x^2)^(3/2))/(48*c^2*e^3) + (g^2*(a + b*x + c*x^2)^(5/2))/(5*c*e) - ((3*b^5*e^5*
g^2 + 256*c^5*d^3*(e*f - d*g)^2 - 384*c^4*d*e*(b*d - a*e)*(e*f - d*g)^2 - 6*b^3*
c*e^4*g*(2*b*e*f - b*d*g + 4*a*e*g) + 16*b*c^2*e^3*(3*a^2*e^2*g^2 + b^2*(e*f - d
*g)^2 + 3*a*b*e*g*(2*e*f - d*g)) + 96*c^3*e^2*(b^2*d*(e*f - d*g)^2 - 2*a*b*e*(e*
f - d*g)^2 - a^2*e^2*g*(2*e*f - d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/(256*c^(7/2)*e^6) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^2*
ArcTanh[(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])])/e^6

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Rubi [A]  time = 3.46123, antiderivative size = 662, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (96 c^3 e^2 \left (-a^2 e^2 g (2 e f-d g)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right )+16 b c^2 e^3 \left (3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right )-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right )}{256 c^{7/2} e^6}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (3 b^2 e^2 g^2-6 c e g x (-b e g-2 c d g+4 c e f)-6 b c e g (2 e f-d g)-16 c^2 (e f-d g)^2\right )}{48 c^2 e^3}+\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (g \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )-6 b^2 c e^3 g (2 a e g-b d g+2 b e f)-32 c^3 e (5 b d-4 a e) (e f-d g)^2+8 b c^2 e^2 \left (3 a e g (2 e f-d g)+2 b (e f-d g)^2\right )+3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2\right )}{128 c^3 e^5}+\frac{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^6}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e} \]

Antiderivative was successfully verified.

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

[Out]

((3*b^4*e^4*g^2 + 128*c^4*d^2*(e*f - d*g)^2 - 32*c^3*e*(5*b*d - 4*a*e)*(e*f - d*
g)^2 - 6*b^2*c*e^3*g*(2*b*e*f - b*d*g + 2*a*e*g) + 8*b*c^2*e^2*(2*b*(e*f - d*g)^
2 + 3*a*e*g*(2*e*f - d*g)) + 2*c*e*((16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a
*e))*g*(4*c*e*f - 2*c*d*g - b*e*g) - 8*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*
x)*Sqrt[a + b*x + c*x^2])/(128*c^3*e^5) - ((3*b^2*e^2*g^2 - 16*c^2*(e*f - d*g)^2
- 6*b*c*e*g*(2*e*f - d*g) - 6*c*e*g*(4*c*e*f - 2*c*d*g - b*e*g)*x)*(a + b*x + c
*x^2)^(3/2))/(48*c^2*e^3) + (g^2*(a + b*x + c*x^2)^(5/2))/(5*c*e) - ((3*b^5*e^5*
g^2 + 256*c^5*d^3*(e*f - d*g)^2 - 384*c^4*d*e*(b*d - a*e)*(e*f - d*g)^2 - 6*b^3*
c*e^4*g*(2*b*e*f - b*d*g + 4*a*e*g) + 16*b*c^2*e^3*(3*a^2*e^2*g^2 + b^2*(e*f - d
*g)^2 + 3*a*b*e*g*(2*e*f - d*g)) + 96*c^3*e^2*(b^2*d*(e*f - d*g)^2 - 2*a*b*e*(e*
f - d*g)^2 - a^2*e^2*g*(2*e*f - d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/(256*c^(7/2)*e^6) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^2*
ArcTanh[(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])])/e^6

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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((g*x+f)**2*(c*x**2+b*x+a)**(3/2)/(e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 3.61516, size = 740, normalized size = 1.12 \[ \frac{\frac{2 e \sqrt{a+x (b+c x)} \left (12 c^2 e^2 \left (32 a^2 e^2 g^2+2 a b e g (-25 d g+50 e f+7 e g x)+b^2 \left (20 d^2 g^2-5 d e g (8 f+g x)+2 e^2 \left (10 f^2+5 f g x+g^2 x^2\right )\right )\right )-30 b^2 c e^3 g (10 a e g+b (-3 d g+6 e f+e g x))+16 c^3 e \left (a e \left (160 d^2 g^2-5 d e g (64 f+15 g x)+2 e^2 \left (80 f^2+75 f g x+24 g^2 x^2\right )\right )+b \left (-150 d^3 g^2+10 d^2 e g (30 f+7 g x)-5 d e^2 \left (30 f^2+28 f g x+9 g^2 x^2\right )+e^3 x \left (70 f^2+90 f g x+33 g^2 x^2\right )\right )\right )+45 b^4 e^4 g^2+32 c^4 \left (60 d^4 g^2-30 d^3 e g (4 f+g x)+20 d^2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )-5 d e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )+2 e^4 x^2 \left (10 f^2+15 f g x+6 g^2 x^2\right )\right )\right )}{c^3}-\frac{15 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (96 c^3 e^2 \left (a^2 e^2 g (d g-2 e f)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right )+16 b c^2 e^3 \left (3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right )-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right )}{c^{7/2}}+3840 (e f-d g)^2 \log (d+e x) \left (e (a e-b d)+c d^2\right )^{3/2}-3840 (e f-d g)^2 \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{3840 e^6} \]

Antiderivative was successfully verified.

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

[Out]

((2*e*Sqrt[a + x*(b + c*x)]*(45*b^4*e^4*g^2 - 30*b^2*c*e^3*g*(10*a*e*g + b*(6*e*
f - 3*d*g + e*g*x)) + 32*c^4*(60*d^4*g^2 - 30*d^3*e*g*(4*f + g*x) + 20*d^2*e^2*(
3*f^2 + 3*f*g*x + g^2*x^2) - 5*d*e^3*x*(6*f^2 + 8*f*g*x + 3*g^2*x^2) + 2*e^4*x^2
*(10*f^2 + 15*f*g*x + 6*g^2*x^2)) + 12*c^2*e^2*(32*a^2*e^2*g^2 + 2*a*b*e*g*(50*e
*f - 25*d*g + 7*e*g*x) + b^2*(20*d^2*g^2 - 5*d*e*g*(8*f + g*x) + 2*e^2*(10*f^2 +
5*f*g*x + g^2*x^2))) + 16*c^3*e*(a*e*(160*d^2*g^2 - 5*d*e*g*(64*f + 15*g*x) + 2
*e^2*(80*f^2 + 75*f*g*x + 24*g^2*x^2)) + b*(-150*d^3*g^2 + 10*d^2*e*g*(30*f + 7*
g*x) - 5*d*e^2*(30*f^2 + 28*f*g*x + 9*g^2*x^2) + e^3*x*(70*f^2 + 90*f*g*x + 33*g
^2*x^2)))))/c^3 + 3840*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*(e*f - d*g)^2*Log[d + e*
x] - (15*(3*b^5*e^5*g^2 + 256*c^5*d^3*(e*f - d*g)^2 - 384*c^4*d*e*(b*d - a*e)*(e
*f - d*g)^2 - 6*b^3*c*e^4*g*(2*b*e*f - b*d*g + 4*a*e*g) + 16*b*c^2*e^3*(3*a^2*e^
2*g^2 + b^2*(e*f - d*g)^2 + 3*a*b*e*g*(2*e*f - d*g)) + 96*c^3*e^2*(b^2*d*(e*f -
d*g)^2 - 2*a*b*e*(e*f - d*g)^2 + a^2*e^2*g*(-2*e*f + d*g)))*Log[b + 2*c*x + 2*Sq
rt[c]*Sqrt[a + x*(b + c*x)]])/c^(7/2) - 3840*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*(e
*f - d*g)^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)]])/(3840*e^6)

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Maple [B]  time = 0.026, size = 6860, normalized size = 10.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError