3.1340 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{\sqrt{b d+2 c d x}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{5 \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{308 c^4 \sqrt{d} \sqrt{a+b x+c x^2}}+\frac{5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{154 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} \sqrt{b d+2 c d x}}{11 c d} \]

[Out]

(5*(b^2 - 4*a*c)^2*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(308*c^3*d) - (5*(
b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2))/(154*c^2*d) + (Sqrt[b*
d + 2*c*d*x]*(a + b*x + c*x^2)^(5/2))/(11*c*d) - (5*(b^2 - 4*a*c)^(13/4)*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])/(308*c^4*Sqrt[d]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.541527, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{5 \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{308 c^4 \sqrt{d} \sqrt{a+b x+c x^2}}+\frac{5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{308 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{154 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} \sqrt{b d+2 c d x}}{11 c d} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b^2 - 4*a*c)^2*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(308*c^3*d) - (5*(
b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2))/(154*c^2*d) + (Sqrt[b*
d + 2*c*d*x]*(a + b*x + c*x^2)^(5/2))/(11*c*d) - (5*(b^2 - 4*a*c)^(13/4)*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])/(308*c^4*Sqrt[d]*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 111.229, size = 218, normalized size = 0.95 \[ \frac{\sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{11 c d} - \frac{5 \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{154 c^{2} d} + \frac{5 \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{308 c^{3} d} - \frac{5 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{308 c^{4} \sqrt{d} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*d + 2*c*d*x)*(a + b*x + c*x**2)**(5/2)/(11*c*d) - 5*(-4*a*c + b**2)*sqrt(
b*d + 2*c*d*x)*(a + b*x + c*x**2)**(3/2)/(154*c**2*d) + 5*(-4*a*c + b**2)**2*sqr
t(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(308*c**3*d) - 5*sqrt(c*(a + b*x + c*x**
2)/(4*a*c - b**2))*(-4*a*c + b**2)**(13/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(
sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(308*c**4*sqrt(d)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.934157, size = 223, normalized size = 0.97 \[ \frac{c (b+2 c x) (a+x (b+c x)) \left (4 c^2 \left (37 a^2+24 a c x^2+7 c^2 x^4\right )+2 b^2 c \left (9 c x^2-25 a\right )+8 b c^2 x \left (12 a+7 c x^2\right )+5 b^4-10 b^3 c x\right )-\frac{5 i \left (b^2-4 a c\right )^3 (b+2 c x)^{3/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{308 c^4 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(b + 2*c*x)*(a + x*(b + c*x))*(5*b^4 - 10*b^3*c*x + 8*b*c^2*x*(12*a + 7*c*x^2
) + 2*b^2*c*(-25*a + 9*c*x^2) + 4*c^2*(37*a^2 + 24*a*c*x^2 + 7*c^2*x^4)) - ((5*I
)*(b^2 - 4*a*c)^3*(b + 2*c*x)^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*El
lipticF[I*ArcSinh[Sqrt[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/Sqrt[-Sqrt[b^2
- 4*a*c]])/(308*c^4*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.03, size = 798, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/616*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)/d*(112*c^7*x^7+392*b*c^6*x^6+496*x
^5*a*c^6+464*x^5*b^2*c^5+320*(-4*a*c+b^2)^(1/2)*((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)*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))*a^3*c^3-240*(-4*a*c+b^2)^(1/2)*((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)*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))*a^2*b
^2*c^2+60*(-4*a*c+b^2)^(1/2)*((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)*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))*a*b^4*c-5*(-4*a*c+b^2)^(1/2)*((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)*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))*b^6+1240*x^4*a*b*c^5+180*x
^4*b^3*c^4+976*x^3*a^2*c^5+752*x^3*a*b^2*c^4-4*x^3*b^4*c^3+1464*x^2*a^2*b*c^4-11
2*a*b^3*c^3*x^2+10*x^2*b^5*c^2+592*a^3*c^4*x+288*a^2*b^2*c^3*x-100*c^2*a*b^4*x+1
0*b^6*c*x+296*a^3*b*c^3-100*a^2*b^3*c^2+10*a*b^5*c)/c^4/(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. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{\sqrt{2 \, c d x + b d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 +
b*x + a)/sqrt(2*c*d*x + b*d), 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((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.12572, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done