3.1323 \(\int \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=236 \[ \frac{\sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{5 c d} \]

[Out]

((b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c*d) - ((b^2 - 4*a*c)^(7/4)*Sqr
t[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])/(5*c^2*Sqrt[a + b*x + c*x^2]) + ((b^
2 - 4*a*c)^(7/4)*Sqrt[d]*Sqrt[-((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])/(5*c^2*Sqrt[a +
b*x + c*x^2])

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Rubi [A]  time = 0.733058, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{5 c d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2],x]

[Out]

((b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c*d) - ((b^2 - 4*a*c)^(7/4)*Sqr
t[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])/(5*c^2*Sqrt[a + b*x + c*x^2]) + ((b^
2 - 4*a*c)^(7/4)*Sqrt[d]*Sqrt[-((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])/(5*c^2*Sqrt[a +
b*x + c*x^2])

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Rubi in Sympy [A]  time = 125.073, size = 221, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{5 c d} - \frac{\sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c^{2} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c^{2} \sqrt{a + b x + c x^{2}}} \]

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)**(1/2),x)

[Out]

(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/(5*c*d) - sqrt(d)*sqrt(c*(a + b*x
+ c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(7/4)*elliptic_e(asin(sqrt(b*d + 2*c*
d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(5*c**2*sqrt(a + b*x + c*x**2)) + sq
rt(d)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(7/4)*elliptic_
f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(5*c**2*sqrt(a
+ b*x + c*x**2))

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Mathematica [C]  time = 2.2166, size = 246, normalized size = 1.04 \[ -\frac{d \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \left (-\frac{c (b+2 c x)^2 (a+x (b+c x))}{\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}-i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{5 c^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2],x]

[Out]

-(d*Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]*(-((c*(b + 2*c*x)^2*(a + x*(b + c*x))
)/Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]) + I*(b^2 - 4*a*c)^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] - I*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Ellipti
cF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(5*c^2*Sqrt[d*(b + 2
*c*x)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.021, size = 492, normalized size = 2.1 \[{\frac{1}{ \left ( 20\,{x}^{3}{c}^{2}+30\,{x}^{2}bc+20\,acx+10\,{b}^{2}x+10\,ab \right ){c}^{2}}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 16\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{c}^{2}-8\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{2}c+\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticE} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ){b}^{4}+8\,{c}^{4}{x}^{4}+16\,b{c}^{3}{x}^{3}+8\,{x}^{2}a{c}^{3}+10\,{x}^{2}{b}^{2}{c}^{2}+8\,xab{c}^{2}+2\,{b}^{3}cx+2\,ac{b}^{2} \right ) } \]

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)^(1/2),x)

[Out]

1/10*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(16*((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^2*c^2-8*((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^2*c+((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^4+8*c^4*x^4+16*b*
c^3*x^3+8*x^2*a*c^3+10*x^2*b^2*c^2+8*x*a*b*c^2+2*b^3*c*x+2*a*c*b^2)/(2*c^2*x^3+3
*b*c*x^2+2*a*c*x+b^2*x+a*b)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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