3.1356 \(\int \frac{1}{(b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=188 \[ \frac{10 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{4 \sqrt{a+b x+c x^2}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}} \]

[Out]

(4*Sqrt[a + b*x + c*x^2])/(7*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(7/2)) + (20*Sqrt[a
+ b*x + c*x^2])/(21*(b^2 - 4*a*c)^2*d^3*(b*d + 2*c*d*x)^(3/2)) + (10*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])/(21*c*(b^2 - 4*a*c)^(7/4)*d^(9/2)*Sqrt[a + b*x + c*
x^2])

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Rubi [A]  time = 0.43327, antiderivative size = 188, 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 \[ \frac{10 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{4 \sqrt{a+b x+c x^2}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(4*Sqrt[a + b*x + c*x^2])/(7*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(7/2)) + (20*Sqrt[a
+ b*x + c*x^2])/(21*(b^2 - 4*a*c)^2*d^3*(b*d + 2*c*d*x)^(3/2)) + (10*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])/(21*c*(b^2 - 4*a*c)^(7/4)*d^(9/2)*Sqrt[a + b*x + c*
x^2])

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Rubi in Sympy [A]  time = 103.925, size = 178, normalized size = 0.95 \[ \frac{4 \sqrt{a + b x + c x^{2}}}{7 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{20 \sqrt{a + b x + c x^{2}}}{21 d^{3} \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{10 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{21 c d^{\frac{9}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

4*sqrt(a + b*x + c*x**2)/(7*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(7/2)) + 20*sqrt(
a + b*x + c*x**2)/(21*d**3*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(3/2)) + 10*sqrt(
c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d
)*(-4*a*c + b**2)**(1/4))), -1)/(21*c*d**(9/2)*(-4*a*c + b**2)**(7/4)*sqrt(a + b
*x + c*x**2))

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Mathematica [C]  time = 0.709183, size = 172, normalized size = 0.91 \[ \frac{2 \left (2 (b+2 c x) (a+x (b+c x)) \left (3 \left (b^2-4 a c\right )+5 (b+2 c x)^2\right )+\frac{5 i (b+2 c x)^{11/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{c \sqrt{-\sqrt{b^2-4 a c}}}\right )}{21 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(2*(2*(b + 2*c*x)*(a + x*(b + c*x))*(3*(b^2 - 4*a*c) + 5*(b + 2*c*x)^2) + ((5*I)
*(b + 2*c*x)^(11/2)*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSin
h[Sqrt[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/(c*Sqrt[-Sqrt[b^2 - 4*a*c]])))
/(21*(b^2 - 4*a*c)^2*(d*(b + 2*c*x))^(9/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.032, size = 691, normalized size = 3.7 \[{\frac{1}{21\,{d}^{5} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) ^{3}c}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 40\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{x}^{3}{c}^{3}+60\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{x}^{2}b{c}^{2}+30\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}x{b}^{2}c+5\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{3}+80\,{c}^{4}{x}^{4}+160\,b{c}^{3}{x}^{3}+32\,{x}^{2}a{c}^{3}+112\,{x}^{2}{b}^{2}{c}^{2}+32\,xab{c}^{2}+32\,{b}^{3}cx-48\,{a}^{2}{c}^{2}+32\,ac{b}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/21*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(40*((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+60*((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+30*((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+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^3+80*c^
4*x^4+160*b*c^3*x^3+32*x^2*a*c^3+112*x^2*b^2*c^2+32*x*a*b*c^2+32*b^3*c*x-48*a^2*
c^2+32*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)/(4*a*c-b^2)^2/(2*c*x
+b)^3/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(9/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(9/2)*sqrt(c*x^2 + b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(9/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

integral(1/((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)*sqrt(c*x^2 + b*x + a)), 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(1/(2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(9/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(9/2)*sqrt(c*x^2 + b*x + a)), x)