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

Optimal. Leaf size=851 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+a} (d+e x)^4}{11 e}+\frac{4 \sqrt{-a} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{4 \sqrt{-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{c x^2+a}}{99 g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{c x^2+a}}{693 c g^4}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{c x^2+a}}{3465 c g^4}-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{c x^2+a}}{3465 c^2 e g^4} \]

[Out]

(-2*(150*a^2*e^4*g^4 - 6*a*c*e^2*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^
2*(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))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3465*c^2*e*g^4) + (2*(d + e*x)^4*Sqrt[
f + g*x]*Sqrt[a + c*x^2])/(11*e) - (2*(2*a*e^2*g^2*(74*e*f - 231*d*g) - c*(233*e
^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3))*(f + g*x)^(3/2)*Sqrt
[a + c*x^2])/(3465*c*g^4) + (2*e*(18*a*e^2*g^2 - c*(29*e^2*f^2 - 96*d*e*f*g + 81
*d^2*g^2))*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(693*c*g^4) + (2*e^2*(e*f - 3*d*g)*(
f + g*x)^(7/2)*Sqrt[a + c*x^2])/(99*g^4) + (4*Sqrt[-a]*(3*a^2*e^2*g^4*(26*e*f +
231*d*g) - c^2*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
) - 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*Sqrt[f
+ g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt
[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3465*c^(3/2)*g^5*Sqrt[(Sqrt[c]*(f +
g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*
(75*a^2*e^3*g^4 - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) - c^2*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))*Sqrt[(Sqrt[c]*(f +
g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (S
qrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3465*c^(5/2
)*g^5*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 4.74988, antiderivative size = 851, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+a} (d+e x)^4}{11 e}+\frac{4 \sqrt{-a} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{4 \sqrt{-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{c x^2+a}}{99 g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{c x^2+a}}{693 c g^4}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{c x^2+a}}{3465 c g^4}-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{c x^2+a}}{3465 c^2 e g^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(150*a^2*e^4*g^4 - 6*a*c*e^2*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^
2*(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))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3465*c^2*e*g^4) + (2*(d + e*x)^4*Sqrt[
f + g*x]*Sqrt[a + c*x^2])/(11*e) - (2*(2*a*e^2*g^2*(74*e*f - 231*d*g) - c*(233*e
^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3))*(f + g*x)^(3/2)*Sqrt
[a + c*x^2])/(3465*c*g^4) + (2*e*(18*a*e^2*g^2 - c*(29*e^2*f^2 - 96*d*e*f*g + 81
*d^2*g^2))*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(693*c*g^4) + (2*e^2*(e*f - 3*d*g)*(
f + g*x)^(7/2)*Sqrt[a + c*x^2])/(99*g^4) + (4*Sqrt[-a]*(3*a^2*e^2*g^4*(26*e*f +
231*d*g) - c^2*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
) - 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*Sqrt[f
+ g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt
[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3465*c^(3/2)*g^5*Sqrt[(Sqrt[c]*(f +
g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*
(75*a^2*e^3*g^4 - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) - c^2*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))*Sqrt[(Sqrt[c]*(f +
g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (S
qrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3465*c^(5/2
)*g^5*Sqrt[f + g*x]*Sqrt[a + 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+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.7258, size = 6884, normalized size = 8.09 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.166, size = 6457, normalized size = 7.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + 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 + a)*sqrt(g*x + f)
, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + c*x**2)*(d + e*x)**3*sqrt(f + g*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError