3.1358 \(\int \frac{(b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{4}{5} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2} \]

[Out]

(4*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/5 + (6*(b^2 - 4*a*c)^(7/4)*d^(
5/2)*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*Sqrt[a + b*x + c*x^2]) - (6*(b^
2 - 4*a*c)^(7/4)*d^(5/2)*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])/(5*c*Sqrt[a + b*
x + c*x^2])

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Rubi [A]  time = 0.725012, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{4}{5} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(5/2)/Sqrt[a + b*x + c*x^2],x]

[Out]

(4*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/5 + (6*(b^2 - 4*a*c)^(7/4)*d^(
5/2)*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*Sqrt[a + b*x + c*x^2]) - (6*(b^
2 - 4*a*c)^(7/4)*d^(5/2)*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])/(5*c*Sqrt[a + b*
x + c*x^2])

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Rubi in Sympy [A]  time = 124.594, size = 221, normalized size = 0.96 \[ \frac{4 d \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{5} + \frac{6 d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c \sqrt{a + b x + c x^{2}}} - \frac{6 d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 2.28953, size = 248, normalized size = 1.07 \[ -\frac{2 d^3 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \left (-\frac{2 c (b+2 c x)^2 (a+x (b+c x))}{\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}+3 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-3 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{5 c \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(5/2)/Sqrt[a + b*x + c*x^2],x]

[Out]

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

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Maple [B]  time = 0.026, size = 496, normalized size = 2.2 \[ -{\frac{{d}^{2}}{5\,c \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 48\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{c}^{2}-24\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{2}c+3\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{4}-16\,{c}^{4}{x}^{4}-32\,b{c}^{3}{x}^{3}-16\,{x}^{2}a{c}^{3}-20\,{x}^{2}{b}^{2}{c}^{2}-16\,xab{c}^{2}-4\,{b}^{3}cx-4\,ac{b}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/5*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^2*(48*((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^2*c^2-24*((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^4-16*c^4*x
^4-32*b*c^3*x^3-16*x^2*a*c^3-20*x^2*b^2*c^2-16*x*a*b*c^2-4*b^3*c*x-4*a*c*b^2)/c/
(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(5/2)/sqrt(c*x^2 + b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

integral((4*c^2*d^2*x^2 + 4*b*c*d^2*x + b^2*d^2)*sqrt(2*c*d*x + b*d)/sqrt(c*x^2
+ b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}}}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d*(b + 2*c*x))**(5/2)/sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(5/2)/sqrt(c*x^2 + b*x + a), x)