3.1328 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{\sqrt{b d+2 c d x}} \, dx\)

Optimal. Leaf size=182 \[ \frac{\left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{14 c^3 \sqrt{d} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{14 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{7 c d} \]

[Out]

-((b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(14*c^2*d) + (Sqrt[b*
d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2))/(7*c*d) + ((b^2 - 4*a*c)^(9/4)*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])/(14*c^3*Sqrt[d]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.431814, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{14 c^3 \sqrt{d} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{14 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{7 c d} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(14*c^2*d) + (Sqrt[b*
d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2))/(7*c*d) + ((b^2 - 4*a*c)^(9/4)*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])/(14*c^3*Sqrt[d]*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 100.35, size = 168, normalized size = 0.92 \[ \frac{\sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{7 c d} - \frac{\left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{14 c^{2} d} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{14 c^{3} \sqrt{d} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*d + 2*c*d*x)*(a + b*x + c*x**2)**(3/2)/(7*c*d) - (-4*a*c + b**2)*sqrt(b*d
+ 2*c*d*x)*sqrt(a + b*x + c*x**2)/(14*c**2*d) + sqrt(c*(a + b*x + c*x**2)/(4*a*
c - b**2))*(-4*a*c + b**2)**(9/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(
-4*a*c + b**2)**(1/4))), -1)/(14*c**3*sqrt(d)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.727229, size = 174, normalized size = 0.96 \[ \frac{c (b+2 c x) (a+x (b+c x)) \left (2 c \left (3 a+c x^2\right )-b^2+2 b c x\right )+\frac{i \left (b^2-4 a c\right )^2 (b+2 c x)^{3/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{14 c^3 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(b + 2*c*x)*(a + x*(b + c*x))*(-b^2 + 2*b*c*x + 2*c*(3*a + c*x^2)) + (I*(b^2
- 4*a*c)^2*(b + 2*c*x)^(3/2)*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]])/(14*c^3*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.031, size = 566, normalized size = 3.1 \[{\frac{1}{28\,d \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ){c}^{3}}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 8\,{c}^{5}{x}^{5}+16\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{a}^{2}{c}^{2}-8\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}a{b}^{2}c+\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}+20\,b{c}^{4}{x}^{4}+32\,{x}^{3}a{c}^{4}+12\,{x}^{3}{b}^{2}{c}^{3}+48\,{x}^{2}ab{c}^{3}-2\,{x}^{2}{b}^{3}{c}^{2}+24\,{a}^{2}{c}^{3}x+12\,a{b}^{2}{c}^{2}x-2\,{b}^{4}cx+12\,{a}^{2}b{c}^{2}-2\,ac{b}^{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)/d*(8*c^5*x^5+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)*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)*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)*E
llipticF(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)*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)*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^4+20*b*c^4*x^4+32*x^3*a*c^4+
12*x^3*b^2*c^3+48*x^2*a*b*c^3-2*x^2*b^3*c^2+24*a^2*c^3*x+12*a*b^2*c^2*x-2*b^4*c*
x+12*a^2*b*c^2-2*a*c*b^3)/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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