3.1361 \(\int \frac{1}{(b d+2 c d x)^{7/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=287 \[ \frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4 \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]

[Out]

(4*Sqrt[a + b*x + c*x^2])/(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + (12*Sqrt[a
+ b*x + c*x^2])/(5*(b^2 - 4*a*c)^2*d^3*Sqrt[b*d + 2*c*d*x]) - (6*Sqrt[-((c*(a +
b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*
c)^(1/4)*Sqrt[d])], -1])/(5*c*(b^2 - 4*a*c)^(5/4)*d^(7/2)*Sqrt[a + b*x + c*x^2])
+ (6*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])/(5*c*(b^2 - 4*a*c)^(5/4)*d^(7/2)*Sq
rt[a + b*x + c*x^2])

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Rubi [A]  time = 0.843015, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4 \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(4*Sqrt[a + b*x + c*x^2])/(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + (12*Sqrt[a
+ b*x + c*x^2])/(5*(b^2 - 4*a*c)^2*d^3*Sqrt[b*d + 2*c*d*x]) - (6*Sqrt[-((c*(a +
b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*
c)^(1/4)*Sqrt[d])], -1])/(5*c*(b^2 - 4*a*c)^(5/4)*d^(7/2)*Sqrt[a + b*x + c*x^2])
+ (6*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])/(5*c*(b^2 - 4*a*c)^(5/4)*d^(7/2)*Sq
rt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 152.422, size = 274, normalized size = 0.95 \[ \frac{4 \sqrt{a + b x + c x^{2}}}{5 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{12 \sqrt{a + b x + c x^{2}}}{5 d^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x}} - \frac{6 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} + \frac{6 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

4*sqrt(a + b*x + c*x**2)/(5*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)) + 12*sqrt(
a + b*x + c*x**2)/(5*d**3*(-4*a*c + b**2)**2*sqrt(b*d + 2*c*d*x)) - 6*sqrt(c*(a
+ b*x + c*x**2)/(4*a*c - b**2))*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4
*a*c + b**2)**(1/4))), -1)/(5*c*d**(7/2)*(-4*a*c + b**2)**(5/4)*sqrt(a + b*x + c
*x**2)) + 6*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)/(5*c*d**(7/2)*(-4*a*c + b**2)**
(5/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.54924, size = 206, normalized size = 0.72 \[ \frac{2 \left (8 (a+x (b+c x)) \left (c \left (3 c x^2-a\right )+b^2+3 b c x\right )-\frac{3 i (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}\right )}{5 d \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(2*(8*(a + x*(b + c*x))*(b^2 + 3*b*c*x + c*(-a + 3*c*x^2)) - ((3*I)*(b + 2*c*x)^
4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(EllipticE[I*ArcSinh[Sqrt[-((b + 2*
c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2
- 4*a*c])]], -1]))/(c*(-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(3/2))))/(5*(b^2 - 4*a
*c)^2*d*(d*(b + 2*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.038, size = 874, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/5*(48*EllipticE(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))*x^2*a*c^3*((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)-12*EllipticE(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))*x^2*b^2*c^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)+48*EllipticE(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))*x*a*b*c^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)-12*EllipticE(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))*x*b^3*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)+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)*EllipticE(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))*a*b^2*c-3*((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)*EllipticE(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))*b^4-48*c^4*x^4-96*b*c
^3*x^3-32*x^2*a*c^3-64*x^2*b^2*c^2-32*x*a*b*c^2-16*b^3*c*x+16*a^2*c^2-16*a*c*b^2
)*(d*(2*c*x+b))^(1/2)/d^4/(c*x^2+b*x+a)^(1/2)/(2*c*x+b)^3/(4*a*c-b^2)^2/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

integral(1/((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)*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)**(7/2)/(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. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)), x)