3.1369 \(\int \frac{(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=162 \[ \frac{20 d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \sqrt{a+b x+c x^2}}+\frac{40}{3} c d^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}-\frac{2 d (b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d*(b*d + 2*c*d*x)^(5/2))/Sqrt[a + b*x + c*x^2] + (40*c*d^3*Sqrt[b*d + 2*c*d*
x]*Sqrt[a + b*x + c*x^2])/3 + (20*(b^2 - 4*a*c)^(5/4)*d^(7/2)*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])/(3*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.405446, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{20 d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \sqrt{a+b x+c x^2}}+\frac{40}{3} c d^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}-\frac{2 d (b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*d*(b*d + 2*c*d*x)^(5/2))/Sqrt[a + b*x + c*x^2] + (40*c*d^3*Sqrt[b*d + 2*c*d*
x]*Sqrt[a + b*x + c*x^2])/3 + (20*(b^2 - 4*a*c)^(5/4)*d^(7/2)*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])/(3*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 91.5556, size = 158, normalized size = 0.98 \[ \frac{40 c d^{3} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{3} + \frac{20 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{5}{2}}}{\sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

40*c*d**3*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/3 + 20*d**(7/2)*sqrt(c*(a +
b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(5/4)*elliptic_f(asin(sqrt(b*d +
2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(3*sqrt(a + b*x + c*x**2)) - 2*
d*(b*d + 2*c*d*x)**(5/2)/sqrt(a + b*x + c*x**2)

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Mathematica [C]  time = 1.06462, size = 175, normalized size = 1.08 \[ -\frac{2 d^3 \sqrt{d (b+2 c x)} \left (\sqrt{-\sqrt{b^2-4 a c}} \left (-4 c \left (5 a+2 c x^2\right )+3 b^2-8 b c x\right )-10 i \left (b^2-4 a c\right ) \sqrt{b+2 c x} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )\right )}{3 \sqrt{-\sqrt{b^2-4 a c}} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*d^3*Sqrt[d*(b + 2*c*x)]*(Sqrt[-Sqrt[b^2 - 4*a*c]]*(3*b^2 - 8*b*c*x - 4*c*(5*
a + 2*c*x^2)) - (10*I)*(b^2 - 4*a*c)*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
]))/(3*Sqrt[-Sqrt[b^2 - 4*a*c]]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.061, size = 366, normalized size = 2.3 \[ -{\frac{2\,{d}^{3}}{6\,{x}^{3}{c}^{2}+9\,{x}^{2}bc+6\,acx+3\,{b}^{2}x+3\,ab}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 20\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}ac-5\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{2}-16\,{c}^{3}{x}^{3}-24\,b{c}^{2}{x}^{2}-40\,a{c}^{2}x-2\,x{b}^{2}c-20\,abc+3\,{b}^{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(3/2),x)

[Out]

-2/3*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(20*((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*c-5*((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^2-16*c^3*x^3-24*b*c^2*x^2-40*a*c^2*x-2*x*b^2*c-20*a*b*c+3*b^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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(7/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(7/2)/(c*x^2 + b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt{2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(7/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((8*c^3*d^3*x^3 + 12*b*c^2*d^3*x^2 + 6*b^2*c*d^3*x + b^3*d^3)*sqrt(2*c*d
*x + b*d)/(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)**(7/2)/(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(7/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(7/2)/(c*x^2 + b*x + a)^(3/2), x)