3.1377 \(\int \frac{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=325 \[ -\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}} \]

[Out]

-2/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2]) - (56*c*Sqrt[a
+ b*x + c*x^2])/(5*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)) - (168*c*Sqrt[a + b*
x + c*x^2])/(5*(b^2 - 4*a*c)^3*d^3*Sqrt[b*d + 2*c*d*x]) + (84*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*(b^2 - 4*a*c)^(9/4)*d^(7/2)*Sqrt[a + b*x + c*x^2]) - (84
*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*(b^2 - 4*a*c)^(9/4)*d^(7/2)*Sqrt[a +
b*x + c*x^2])

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Rubi [A]  time = 0.954374, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2]) - (56*c*Sqrt[a
+ b*x + c*x^2])/(5*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)) - (168*c*Sqrt[a + b*
x + c*x^2])/(5*(b^2 - 4*a*c)^3*d^3*Sqrt[b*d + 2*c*d*x]) + (84*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*(b^2 - 4*a*c)^(9/4)*d^(7/2)*Sqrt[a + b*x + c*x^2]) - (84
*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*(b^2 - 4*a*c)^(9/4)*d^(7/2)*Sqrt[a +
b*x + c*x^2])

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Rubi in Sympy [A]  time = 171.66, size = 314, normalized size = 0.97 \[ - \frac{56 c \sqrt{a + b x + c x^{2}}}{5 d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{168 c \sqrt{a + b x + c x^{2}}}{5 d^{3} \left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x}} - \frac{2}{d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}} + \frac{84 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \sqrt{a + b x + c x^{2}}} - \frac{84 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \sqrt{a + b x + c x^{2}}} \]

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)**(3/2),x)

[Out]

-56*c*sqrt(a + b*x + c*x**2)/(5*d*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(5/2)) - 1
68*c*sqrt(a + b*x + c*x**2)/(5*d**3*(-4*a*c + b**2)**3*sqrt(b*d + 2*c*d*x)) - 2/
(d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)) + 84*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*d**(7/2)*(-4*a*c + b**2)**(9/4)*sqrt(a + b*x + c*
x**2)) - 84*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*d**(7/2)*(-4*a*c + b**2)**(9
/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 2.14071, size = 221, normalized size = 0.68 \[ \frac{2 \left (-8 c \left (b^2-4 a c\right ) (a+x (b+c x))+\frac{42 i (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}-64 c (b+2 c x)^2 (a+x (b+c x))-5 (b+2 c x)^4\right )}{5 d \left (b^2-4 a c\right )^3 \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(2*(-5*(b + 2*c*x)^4 - 8*c*(b^2 - 4*a*c)*(a + x*(b + c*x)) - 64*c*(b + 2*c*x)^2*
(a + x*(b + c*x)) + ((42*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] - Ellipti
cF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(-((b + 2*c*x)/Sqrt[
b^2 - 4*a*c]))^(3/2)))/(5*(b^2 - 4*a*c)^3*d*(d*(b + 2*c*x))^(5/2)*Sqrt[a + x*(b
+ c*x)])

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Maple [B]  time = 0.043, size = 876, normalized size = 2.7 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

-2/5*(336*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)-84*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)+336*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)-84*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)+84*((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-21*((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^4-336*c^4*x^4-67
2*b*c^3*x^3-224*x^2*a*c^3-448*x^2*b^2*c^2-224*x*a*b*c^2-112*b^3*c*x+32*a^2*c^2-7
2*a*c*b^2-5*b^4)*(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)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((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. \[{\rm integral}\left (\frac{1}{{\left (8 \, c^{4} d^{3} x^{5} + 20 \, b c^{3} d^{3} x^{4} + a b^{3} d^{3} + 2 \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{3} +{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{2} +{\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((8*c^4*d^3*x^5 + 20*b*c^3*d^3*x^4 + a*b^3*d^3 + 2*(9*b^2*c^2 + 4*a*c
^3)*d^3*x^3 + (7*b^3*c + 12*a*b*c^2)*d^3*x^2 + (b^4 + 6*a*b^2*c)*d^3*x)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2)), x)