3.1330 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{14 c^3 d^{9/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(14*c^2*d^3*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3
/2)/(7*c*d*(b*d + 2*c*d*x)^(7/2)) + ((b^2 - 4*a*c)^(1/4)*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*d^(9/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.419633, antiderivative size = 174, 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 \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{14 c^3 d^{9/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(9/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(14*c^2*d^3*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3
/2)/(7*c*d*(b*d + 2*c*d*x)^(7/2)) + ((b^2 - 4*a*c)^(1/4)*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*d^(9/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 95.3999, size = 162, normalized size = 0.93 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{7 c d \left (b d + 2 c d x\right )^{\frac{7}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{14 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt [4]{- 4 a c + b^{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{14 c^{3} d^{\frac{9}{2}} \sqrt{a + b x + c x^{2}}} \]

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)**(9/2),x)

[Out]

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

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Mathematica [C]  time = 0.660784, size = 159, normalized size = 0.91 \[ \frac{c (b+2 c x) (a+x (b+c x)) \left (-4 a c+b^2-3 (b+2 c x)^2\right )+\frac{2 i (b+2 c x)^{11/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{28 c^3 \sqrt{a+x (b+c x)} (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(c*(b + 2*c*x)*(a + x*(b + c*x))*(b^2 - 4*a*c - 3*(b + 2*c*x)^2) + ((2*I)*(b + 2
*c*x)^(11/2)*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]])/(28*c^3*(d*
(b + 2*c*x))^(9/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.038, size = 678, normalized size = 3.9 \[{\frac{1}{28\,{d}^{5} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ){c}^{3} \left ( 2\,cx+b \right ) ^{3}}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 8\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{x}^{3}{c}^{3}+12\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{x}^{2}b{c}^{2}+6\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}x{b}^{2}c+\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{3}-12\,{c}^{4}{x}^{4}-24\,b{c}^{3}{x}^{3}-16\,{x}^{2}a{c}^{3}-14\,{x}^{2}{b}^{2}{c}^{2}-16\,xab{c}^{2}-2\,{b}^{3}cx-4\,{a}^{2}{c}^{2}-2\,ac{b}^{2} \right ) } \]

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)^(9/2),x)

[Out]

1/28*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(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)*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)*x^3*c^3+12*((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)*x^2*b*c^2+6*((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)*x*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^3-12*c^4*x^
4-24*b*c^3*x^3-16*x^2*a*c^3-14*x^2*b^2*c^2-16*x*a*b*c^2-2*b^3*c*x-4*a^2*c^2-2*a*
c*b^2)/d^5/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/c^3/(2*c*x+b)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(9/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/2)/((16*c^4*d^4*x^4 + 32*b*c^3*d^4*x^3 + 24*b^2*c^
2*d^4*x^2 + 8*b^3*c*d^4*x + b^4*d^4)*sqrt(2*c*d*x + b*d)), 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((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(9/2), x)