3.886 \(\int (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=1551 \[ \text{result too large to display} \]

[Out]

(-2*(64*b^4*e^4*g^4 + 4*b^2*c*e^3*g^3*(7*b*e*f - 66*b*d*g - 69*a*e*g) + c^4*(187
*e^4*f^4 - 732*d*e^3*f^3*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^
4) + 3*c^2*e^2*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*g) + 3*b^2*(e^2*f^2
- 11*d*e*f*g + 44*d^2*g^2)) - c^3*e*g*(6*a*e*g*(2*e^2*f^2 - 33*d*e*f*g + 165*d^
2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^3)))*Sqrt[f + g*x]*Sqrt[a + b
*x + c*x^2])/(3465*c^4*e*g^4) + (2*(d + e*x)^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^
2])/(11*e) + (2*(48*b^3*e^3*g^3 + b*c*e^2*g^2*(67*b*e*f - 198*b*d*g - 157*a*e*g)
+ c^3*(233*e^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3) - c^2*e*
g*(2*a*e*g*(74*e*f - 231*d*g) - 3*b*(24*e^2*f^2 - 88*d*e*f*g + 99*d^2*g^2)))*(f
+ g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(3465*c^3*g^4) - (2*e*(8*b^2*e^2*g^2 + c*e*g
*(19*b*e*f - 33*b*d*g - 18*a*e*g) + c^2*(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*
(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(693*c^2*g^4) + (2*e^2*(c*e*f - 3*c*d*g +
b*e*g)*(f + g*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(99*c*g^4) + (Sqrt[2]*Sqrt[b^2 -
4*a*c]*(128*b^5*e^3*g^5 - 8*b^3*c*e^2*g^4*(7*b*e*f + 66*b*d*g + 87*a*e*g) + 2*c^
5*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + b*c^2*e*g
^3*(771*a^2*e^2*g^2 + 6*a*b*e*g*(43*e*f + 396*d*g) - b^2*(37*e^2*f^2 - 264*d*e*f
*g - 792*d^2*g^2)) - c^4*g*(b*f*(56*e^3*f^3 - 264*d*e^2*f^2*g + 495*d^2*e*f*g^2
- 462*d^3*g^3) - 18*a*g*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^
3)) - c^3*g^2*(6*a^2*e^2*g^2*(26*e*f + 231*d*g) - 9*a*b*e*g*(15*e^2*f^2 - 110*d*
e*f*g - 319*d^2*g^2) + b^2*(37*e^3*f^3 - 198*d*e^2*f^2*g + 495*d^2*e*f*g^2 + 462
*d^3*g^3)))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE
[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq
rt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(3465*c^5*g^5*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)*(64*b^4*e^3*g^4 + 4*b^2*c*e^2*g^3*(
7*b*e*f - 66*b*d*g - 69*a*e*g) - 2*c^4*f*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2
*e*f*g^2 - 231*d^3*g^3) + 3*c^2*e*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*
g) + 3*b^2*(e^2*f^2 - 11*d*e*f*g + 44*d^2*g^2)) - c^3*g*(6*a*e*g*(2*e^2*f^2 - 33
*d*e*f*g + 165*d^2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^3)))*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)])/
(3465*c^5*g^5*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 17.5545, antiderivative size = 1551, normalized size of antiderivative = 1., number of steps used = 10, 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)^4}{11 e}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\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 )}{3465 c^5 g^5 \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 (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^4-g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\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 )}{3465 c^5 g^5 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt{c x^2+b x+a}}{99 c g^4}-\frac{2 e \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{693 c^2 g^4}+\frac{2 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{3465 c^3 g^4}-\frac{2 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{3465 c^4 e g^4} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*(64*b^4*e^4*g^4 + 4*b^2*c*e^3*g^3*(7*b*e*f - 66*b*d*g - 69*a*e*g) + c^4*(187
*e^4*f^4 - 732*d*e^3*f^3*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^
4) + 3*c^2*e^2*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*g) + 3*b^2*(e^2*f^2
- 11*d*e*f*g + 44*d^2*g^2)) - c^3*e*g*(6*a*e*g*(2*e^2*f^2 - 33*d*e*f*g + 165*d^
2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^3)))*Sqrt[f + g*x]*Sqrt[a + b
*x + c*x^2])/(3465*c^4*e*g^4) + (2*(d + e*x)^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^
2])/(11*e) + (2*(48*b^3*e^3*g^3 + b*c*e^2*g^2*(67*b*e*f - 198*b*d*g - 157*a*e*g)
+ c^3*(233*e^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3) - c^2*e*
g*(2*a*e*g*(74*e*f - 231*d*g) - 3*b*(24*e^2*f^2 - 88*d*e*f*g + 99*d^2*g^2)))*(f
+ g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(3465*c^3*g^4) - (2*e*(8*b^2*e^2*g^2 + c*e*g
*(19*b*e*f - 33*b*d*g - 18*a*e*g) + c^2*(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*
(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(693*c^2*g^4) + (2*e^2*(c*e*f - 3*c*d*g +
b*e*g)*(f + g*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(99*c*g^4) + (Sqrt[2]*Sqrt[b^2 -
4*a*c]*(128*b^5*e^3*g^5 - 8*b^3*c*e^2*g^4*(7*b*e*f + 66*b*d*g + 87*a*e*g) + 2*c^
5*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + b*c^2*e*g
^3*(771*a^2*e^2*g^2 + 6*a*b*e*g*(43*e*f + 396*d*g) - b^2*(37*e^2*f^2 - 264*d*e*f
*g - 792*d^2*g^2)) - c^4*g*(b*f*(56*e^3*f^3 - 264*d*e^2*f^2*g + 495*d^2*e*f*g^2
- 462*d^3*g^3) - 18*a*g*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^
3)) - c^3*g^2*(6*a^2*e^2*g^2*(26*e*f + 231*d*g) - 9*a*b*e*g*(15*e^2*f^2 - 110*d*
e*f*g - 319*d^2*g^2) + b^2*(37*e^3*f^3 - 198*d*e^2*f^2*g + 495*d^2*e*f*g^2 + 462
*d^3*g^3)))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE
[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq
rt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(3465*c^5*g^5*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)*(64*b^4*e^3*g^4 + 4*b^2*c*e^2*g^3*(
7*b*e*f - 66*b*d*g - 69*a*e*g) - 2*c^4*f*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2
*e*f*g^2 - 231*d^3*g^3) + 3*c^2*e*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*
g) + 3*b^2*(e^2*f^2 - 11*d*e*f*g + 44*d^2*g^2)) - c^3*g*(6*a*e*g*(2*e^2*f^2 - 33
*d*e*f*g + 165*d^2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^3)))*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)])/
(3465*c^5*g^5*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)**3*(g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 22.3229, size = 32331, normalized size = 20.85 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^3*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 = 0.239, size = 32647, normalized size = 21.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(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. \[ \int \sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{3} \sqrt{g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3*sqrt(g*x + f),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3*sqrt(g*x + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + b x + a} \sqrt{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*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. \[ \int \left (d + e x\right )^{3} \sqrt{f + g x} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d + e*x)**3*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)^3*sqrt(g*x + f),x, algorithm="giac")

[Out]

Timed out