3.1334 \(\int \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=281 \[ -\frac{\sqrt{d} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 \sqrt{a+b x+c x^2}}+\frac{\sqrt{d} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{30 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{9 c d} \]

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(30*c^2*d) + ((b*d
+ 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c*d) + ((b^2 - 4*a*c)^(11/4)*Sqrt[d
]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d
*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*Sqrt[a + b*x + c*x^2]) - ((b^2
- 4*a*c)^(11/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[A
rcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*Sqrt[a +
b*x + c*x^2])

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Rubi [A]  time = 0.830126, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{d} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 \sqrt{a+b x+c x^2}}+\frac{\sqrt{d} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{30 c^3 \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{30 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{9 c d} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(30*c^2*d) + ((b*d
+ 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c*d) + ((b^2 - 4*a*c)^(11/4)*Sqrt[d
]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d
*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*Sqrt[a + b*x + c*x^2]) - ((b^2
- 4*a*c)^(11/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[A
rcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*Sqrt[a +
b*x + c*x^2])

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Rubi in Sympy [A]  time = 147.516, size = 264, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{9 c d} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{30 c^{2} d} + \frac{\sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{30 c^{3} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{30 c^{3} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

(b*d + 2*c*d*x)**(3/2)*(a + b*x + c*x**2)**(3/2)/(9*c*d) - (-4*a*c + b**2)*(b*d
+ 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/(30*c**2*d) + sqrt(d)*sqrt(c*(a + b*x +
c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(11/4)*elliptic_e(asin(sqrt(b*d + 2*c*
d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(30*c**3*sqrt(a + b*x + c*x**2)) - s
qrt(d)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(11/4)*ellipti
c_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(30*c**3*sqr
t(a + b*x + c*x**2))

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Mathematica [C]  time = 1.03325, size = 206, normalized size = 0.73 \[ \frac{\sqrt{d (b+2 c x)} \left (c (b+2 c x) (a+x (b+c x)) \left (2 c \left (11 a+5 c x^2\right )-3 b^2+10 b c x\right )-\frac{3 i \left (b^2-4 a c\right )^{5/2} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}\right )}{90 c^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d*(b + 2*c*x)]*(c*(b + 2*c*x)*(a + x*(b + c*x))*(-3*b^2 + 10*b*c*x + 2*c*(
11*a + 5*c*x^2)) - ((3*I)*(b^2 - 4*a*c)^(5/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 +
4*a*c)]*(EllipticE[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - Ell
ipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/Sqrt[-((b + 2*c*
x)/Sqrt[b^2 - 4*a*c])]))/(90*c^3*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.026, size = 700, normalized size = 2.5 \[{\frac{1}{180\,{c}^{3} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 80\,{c}^{6}{x}^{6}+240\,b{c}^{5}{x}^{5}+192\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{3}{c}^{3}-144\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{b}^{2}{c}^{2}+36\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{4}c-3\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{6}+256\,{x}^{4}a{c}^{5}+236\,{x}^{4}{b}^{2}{c}^{4}+512\,{x}^{3}ab{c}^{4}+72\,{b}^{3}{c}^{3}{x}^{3}+176\,{x}^{2}{a}^{2}{c}^{4}+296\,{x}^{2}a{b}^{2}{c}^{3}-10\,{x}^{2}{b}^{4}{c}^{2}+176\,{a}^{2}b{c}^{3}x+40\,a{b}^{3}{c}^{2}x-6\,{b}^{5}cx+44\,{a}^{2}{b}^{2}{c}^{2}-6\,a{b}^{4}c \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/180*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(80*c^6*x^6+240*b*c^5*x^5+192*((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)*EllipticE(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^
3-144*((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)*Ell
ipticE(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+36*((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)*EllipticE(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-3*((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)*EllipticE(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+256*x^4*a*c^5+236*x^4*b^2*c^4+512*x^3*a*b*c^4+72*b^
3*c^3*x^3+176*x^2*a^2*c^4+296*x^2*a*b^2*c^3-10*x^2*b^4*c^2+176*a^2*b*c^3*x+40*a*
b^3*c^2*x-6*b^5*c*x+44*a^2*b^2*c^2-6*a*b^4*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. \[ \int \sqrt{2 \, c d x + b d}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 20.1743, size = 264, normalized size = 0.94 \[ \frac{a \left (b d + 2 c d x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname{polar\_lift}{\left (a - \frac{b^{2}}{4 c} \right )}}} \right )} \sqrt{\operatorname{polar\_lift}{\left (a - \frac{b^{2}}{4 c} \right )}}}{4 c d \Gamma \left (\frac{7}{4}\right )} - \frac{b^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname{polar\_lift}{\left (a - \frac{b^{2}}{4 c} \right )}}} \right )} \sqrt{\operatorname{polar\_lift}{\left (a - \frac{b^{2}}{4 c} \right )}}}{16 c^{2} d \Gamma \left (\frac{7}{4}\right )} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname{polar\_lift}{\left (a - \frac{b^{2}}{4 c} \right )}}} \right )} \sqrt{\operatorname{polar\_lift}{\left (a - \frac{b^{2}}{4 c} \right )}}}{16 c^{2} d^{3} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), (b*d + 2*c*d*x)**
2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**
2/(4*c)))/(4*c*d*gamma(7/4)) - b**2*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper((-1/
2, 3/4), (7/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b**
2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(16*c**2*d*gamma(7/4)) + (b*d + 2*c*
d*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), (b*d + 2*c*d*x)**2*exp_polar(
I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(1
6*c**2*d**3*gamma(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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