3.1327 \(\int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=227 \[ \frac{d^{3/2} \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{154 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{77 c^2}-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{154 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d} \]

[Out]

((b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(77*c^2) - (3*(b^2
- 4*a*c)*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(154*c^2*d) + ((b*d + 2*c
*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(11*c*d) + ((b^2 - 4*a*c)^(13/4)*d^(3/2)*Sq
rt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/
((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(154*c^3*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.5307, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{d^{3/2} \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{154 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{77 c^2}-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{154 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(77*c^2) - (3*(b^2
- 4*a*c)*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(154*c^2*d) + ((b*d + 2*c
*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(11*c*d) + ((b^2 - 4*a*c)^(13/4)*d^(3/2)*Sq
rt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/
((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(154*c^3*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 111.309, size = 214, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{11 c d} + \frac{d \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{77 c^{2}} - \frac{3 \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{154 c^{2} d} + \frac{d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{154 c^{3} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(3/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

(b*d + 2*c*d*x)**(5/2)*(a + b*x + c*x**2)**(3/2)/(11*c*d) + d*(-4*a*c + b**2)**2
*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(77*c**2) - 3*(-4*a*c + b**2)*(b*d +
2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/(154*c**2*d) + d**(3/2)*sqrt(c*(a + b*x
+ c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(13/4)*elliptic_f(asin(sqrt(b*d + 2*c
*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(154*c**3*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.06575, size = 225, normalized size = 0.99 \[ \frac{(d (b+2 c x))^{3/2} \left (\frac{c (a+x (b+c x)) \left (8 c^2 \left (4 a^2+13 a c x^2+7 c^2 x^4\right )+2 b^2 c \left (5 a+29 c x^2\right )+8 b c^2 x \left (13 a+14 c x^2\right )-b^4+2 b^3 c x\right )}{b+2 c x}+\frac{i \left (b^2-4 a c\right )^3 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} \sqrt{b+2 c x}}\right )}{154 c^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(3/2)*((c*(a + x*(b + c*x))*(-b^4 + 2*b^3*c*x + 8*b*c^2*x*(13*a
+ 14*c*x^2) + 2*b^2*c*(5*a + 29*c*x^2) + 8*c^2*(4*a^2 + 13*a*c*x^2 + 7*c^2*x^4)
))/(b + 2*c*x) + (I*(b^2 - 4*a*c)^3*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*El
lipticF[I*ArcSinh[Sqrt[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/(Sqrt[-Sqrt[b^
2 - 4*a*c]]*Sqrt[b + 2*c*x])))/(154*c^3*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.029, size = 796, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(3/2),x)

[Out]

-1/308*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d*(-224*c^7*x^7-784*b*c^6*x^6-640
*x^5*a*c^6-1016*x^5*b^2*c^5+64*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a
*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1
/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^3*c^3-48*(-4*a*c+b^2)^(1/2)*((
b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(
1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1
/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^2*
b^2*c^2+12*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^
(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a
*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1
/2))^(1/2)*2^(1/2),2^(1/2))*a*b^4*c-(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1
/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+
(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^6-1600*x^4*a*b*c^5-580*x^
4*b^3*c^4-544*x^3*a^2*c^5-1328*x^3*a*b^2*c^4-124*x^3*b^4*c^3-816*x^2*a^2*b*c^4-3
92*a*b^3*c^3*x^2+2*x^2*b^5*c^2-128*a^3*c^4*x-312*a^2*b^2*c^3*x-20*c^2*a*b^4*x+2*
b^6*c*x-64*a^3*b*c^3-20*a^2*b^3*c^2+2*a*b^5*c)/c^3/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+
b^2*x+a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((2*c^2*d*x^3 + 3*b*c*d*x^2 + a*b*d + (b^2 + 2*a*c)*d*x)*sqrt(2*c*d*x +
b*d)*sqrt(c*x^2 + b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*(b + 2*c*x))**(3/2)*(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 1.01088, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done