3.1370 \(\int \frac{(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=125 \[ \frac{4 d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d*Sqrt[b*d + 2*c*d*x])/Sqrt[a + b*x + c*x^2] + (4*(b^2 - 4*a*c)^(1/4)*d^(3/2
)*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])/Sqrt[a + b*x + c*x^2]

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Rubi [A]  time = 0.332587, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4 d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*Sqrt[b*d + 2*c*d*x])/Sqrt[a + b*x + c*x^2] + (4*(b^2 - 4*a*c)^(1/4)*d^(3/2
)*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])/Sqrt[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 83.8829, size = 121, normalized size = 0.97 \[ \frac{4 d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt [4]{- 4 a c + b^{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt{a + b x + c x^{2}}} - \frac{2 d \sqrt{b d + 2 c d x}}{\sqrt{a + b x + c x^{2}}} \]

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]

4*d**(3/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(1/4)*elli
ptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/sqrt(a +
b*x + c*x**2) - 2*d*sqrt(b*d + 2*c*d*x)/sqrt(a + b*x + c*x**2)

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Mathematica [C]  time = 0.552894, size = 138, normalized size = 1.1 \[ -\frac{2 d \sqrt{d (b+2 c x)} \left (\sqrt{-\sqrt{b^2-4 a c}}-2 i \sqrt{b+2 c x} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )\right )}{\sqrt{-\sqrt{b^2-4 a c}} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.053, size = 194, normalized size = 1.6 \[ 2\,{\frac{\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a}d}{2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab} \left ({\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-2\,cx-b \right ) } \]

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]

2*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d*(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))*(-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)-2*c*x-b)/(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

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. \[{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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

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="giac")

[Out]

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