3.887 \(\int (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=1015 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^4+g \left (3 a g \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e g f+21 d^2 g^2\right )\right ) c^3+3 g^2 \left (-\left (e^2 f^2-5 d e g f-7 d^2 g^2\right ) b^2+a e g (5 e f+29 d g) b+7 a^2 e^2 g^2\right ) c^2-4 b^2 e g^3 (b e f+6 b d g+9 a e g) c+8 b^4 e^2 g^4\right ) \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^3-3 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^2 g^3\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^3}-\frac{4 \left (\left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^2+e g (4 b e f-9 b d g-7 a e g) c+3 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^3}+\frac{2 \left (\left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right ) c^3-3 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e^2 g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{315 c^3 e g^3} \]

[Out]

(2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) + c^3*(19*e^3*f^3
- 57*d*e^2*f^2*g + 63*d^2*e*f*g^2 - 35*d^3*g^3) - 3*c^2*e*g^2*(2*a*e*(e*f - 10*d
*g) + b*d*(2*e*f - 7*d*g)))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(315*c^3*e*g^3)
+ (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(9*e) - (4*(3*b^2*e^2*g^2
+ c*e*g*(4*b*e*f - 9*b*d*g - 7*a*e*g) + c^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^
2))*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(315*c^2*g^3) + (2*e*(c*e*f - 3*c*d*g
+ b*e*g)*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(63*c*g^3) - (2*Sqrt[2]*Sqrt[b^
2 - 4*a*c]*(8*b^4*e^2*g^4 - 4*b^2*c*e*g^3*(b*e*f + 6*b*d*g + 9*a*e*g) + c^4*f^2*
(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) + 3*c^2*g^2*(7*a^2*e^2*g^2 + a*b*e*g*(5*e*
f + 29*d*g) - b^2*(e^2*f^2 - 5*d*e*f*g - 7*d^2*g^2)) + c^3*g*(3*a*g*(3*e^2*f^2 -
16*d*e*f*g - 21*d^2*g^2) - b*f*(4*e^2*f^2 - 15*d*e*f*g + 21*d^2*g^2)))*Sqrt[f +
g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sq
rt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(
2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(315*c^4*g^4*Sqrt[(c*(f + g*x))/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*
(c*f^2 - b*f*g + a*g^2)*(8*b^3*e^2*g^3 + 3*b*c*e*g^2*(b*e*f - 8*b*d*g - 9*a*e*g)
- 2*c^3*f*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*c^2*g^2*(2*a*e*(e*f - 10*d*
g) + b*d*(2*e*f - 7*d*g)))*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g
)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[
b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c
*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(315*c^4*g^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^
2])

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Rubi [A]  time = 8.27682, antiderivative size = 1015, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^4+g \left (3 a g \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e g f+21 d^2 g^2\right )\right ) c^3+3 g^2 \left (-\left (e^2 f^2-5 d e g f-7 d^2 g^2\right ) b^2+a e g (5 e f+29 d g) b+7 a^2 e^2 g^2\right ) c^2-4 b^2 e g^3 (b e f+6 b d g+9 a e g) c+8 b^4 e^2 g^4\right ) \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^3-3 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^2 g^3\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^3}-\frac{4 \left (\left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^2+e g (4 b e f-9 b d g-7 a e g) c+3 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^3}+\frac{2 \left (\left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right ) c^3-3 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e^2 g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{315 c^3 e g^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) + c^3*(19*e^3*f^3
- 57*d*e^2*f^2*g + 63*d^2*e*f*g^2 - 35*d^3*g^3) - 3*c^2*e*g^2*(2*a*e*(e*f - 10*d
*g) + b*d*(2*e*f - 7*d*g)))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(315*c^3*e*g^3)
+ (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(9*e) - (4*(3*b^2*e^2*g^2
+ c*e*g*(4*b*e*f - 9*b*d*g - 7*a*e*g) + c^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^
2))*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(315*c^2*g^3) + (2*e*(c*e*f - 3*c*d*g
+ b*e*g)*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(63*c*g^3) - (2*Sqrt[2]*Sqrt[b^
2 - 4*a*c]*(8*b^4*e^2*g^4 - 4*b^2*c*e*g^3*(b*e*f + 6*b*d*g + 9*a*e*g) + c^4*f^2*
(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) + 3*c^2*g^2*(7*a^2*e^2*g^2 + a*b*e*g*(5*e*
f + 29*d*g) - b^2*(e^2*f^2 - 5*d*e*f*g - 7*d^2*g^2)) + c^3*g*(3*a*g*(3*e^2*f^2 -
16*d*e*f*g - 21*d^2*g^2) - b*f*(4*e^2*f^2 - 15*d*e*f*g + 21*d^2*g^2)))*Sqrt[f +
g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sq
rt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(
2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(315*c^4*g^4*Sqrt[(c*(f + g*x))/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*
(c*f^2 - b*f*g + a*g^2)*(8*b^3*e^2*g^3 + 3*b*c*e*g^2*(b*e*f - 8*b*d*g - 9*a*e*g)
- 2*c^3*f*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*c^2*g^2*(2*a*e*(e*f - 10*d*
g) + b*d*(2*e*f - 7*d*g)))*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g
)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[
b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c
*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(315*c^4*g^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^
2])

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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((e*x+d)**2*(g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 18.2186, size = 15781, normalized size = 15.55 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.078, size = 20224, normalized size = 19.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{2} \sqrt{g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^2*sqrt(g*x + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^2*x^2 + 2*d*e*x + d^2)*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{2} \sqrt{f + g x} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d + e*x)**2*sqrt(f + g*x)*sqrt(a + b*x + c*x**2), x)

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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(sqrt(c*x^2 + b*x + a)*(e*x + d)^2*sqrt(g*x + f),x, algorithm="giac")

[Out]

Timed out