3.1333 \(\int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=326 \[ -\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{13 c d} \]

[Out]

((b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(195*c^2) - ((b^
2 - 4*a*c)*(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(78*c^2*d) + ((b*d + 2*c
*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2))/(13*c*d) + ((b^2 - 4*a*c)^(15/4)*d^(5/2)*Sq
rt[-((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])/(130*c^3*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4
*a*c)^(15/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcS
in[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(130*c^3*Sqrt[a + b*
x + c*x^2])

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Rubi [A]  time = 0.989742, antiderivative size = 326, 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 \[ -\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{13 c d} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2),x]

[Out]

((b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(195*c^2) - ((b^
2 - 4*a*c)*(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(78*c^2*d) + ((b*d + 2*c
*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2))/(13*c*d) + ((b^2 - 4*a*c)^(15/4)*d^(5/2)*Sq
rt[-((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])/(130*c^3*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4
*a*c)^(15/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcS
in[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(130*c^3*Sqrt[a + b*
x + c*x^2])

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Rubi in Sympy [A]  time = 165.259, size = 308, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{13 c d} + \frac{d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{195 c^{2}} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}}{78 c^{2} d} + \frac{d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{130 c^{3} \sqrt{a + b x + c x^{2}}} - \frac{d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{130 c^{3} \sqrt{a + b x + c x^{2}}} \]

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)**(3/2),x)

[Out]

(b*d + 2*c*d*x)**(7/2)*(a + b*x + c*x**2)**(3/2)/(13*c*d) + d*(-4*a*c + b**2)**2
*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/(195*c**2) - (-4*a*c + b**2)*(b*d
+ 2*c*d*x)**(7/2)*sqrt(a + b*x + c*x**2)/(78*c**2*d) + d**(5/2)*sqrt(c*(a + b*x
+ c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(15/4)*elliptic_e(asin(sqrt(b*d + 2*
c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(130*c**3*sqrt(a + b*x + c*x**2))
- d**(5/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(15/4)*ell
iptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(130*c**
3*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.83277, size = 256, normalized size = 0.79 \[ \frac{(d (b+2 c x))^{5/2} \left (\frac{c (a+x (b+c x)) \left (8 c^2 \left (4 a^2+25 a c x^2+15 c^2 x^4\right )+2 b^2 c \left (17 a+65 c x^2\right )+40 b c^2 x \left (5 a+6 c x^2\right )-3 b^4+10 b^3 c x\right )}{b+2 c x}-\frac{3 i \left (b^2-4 a c\right )^{5/2} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{5/2}}\right )}{390 c^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2),x]

[Out]

((d*(b + 2*c*x))^(5/2)*((c*(a + x*(b + c*x))*(-3*b^4 + 10*b^3*c*x + 40*b*c^2*x*(
5*a + 6*c*x^2) + 2*b^2*c*(17*a + 65*c*x^2) + 8*c^2*(4*a^2 + 25*a*c*x^2 + 15*c^2*
x^4)))/(b + 2*c*x) - ((3*I)*(b^2 - 4*a*c)^(5/2)*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] - E
llipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(-((b + 2*c*x)
/Sqrt[b^2 - 4*a*c]))^(5/2)))/(390*c^3*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.032, size = 938, normalized size = 2.9 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

-1/780*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^2*(-2560*x^6*a*c^7-1856*x^4*a^2
*c^6-256*x^2*a^3*c^5-4800*x^5*b^3*c^5+10*x^2*b^6*c^2-1916*x^4*b^4*c^4+6*x*b^7*c-
3840*x^7*b*c^7-6080*x^6*b^2*c^6-312*x^3*b^5*c^3+768*((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^4*c^4-960*x^8*c^8-64*a^3*b^2
*c^3-68*a^2*b^4*c^2+6*a*b^6*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^8-768*((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^3*b^2*c^3+288*((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*b^4*c^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*b^6*c-7680*x^5*
a*b*c^6-8672*x^4*a*b^2*c^5-3712*x^3*a^2*b*c^5-4544*x^3*a*b^3*c^4-2592*x^2*a^2*b^
2*c^4-1056*x^2*a*b^4*c^3-256*x*a^3*b*c^4-736*x*a^2*b^3*c^3-64*x*a*b^5*c^2)/c^3/(
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. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (4 \, c^{3} d^{2} x^{4} + 8 \, b c^{2} d^{2} x^{3} + a b^{2} d^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} + 4 \, a b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((4*c^3*d^2*x^4 + 8*b*c^2*d^2*x^3 + a*b^2*d^2 + (5*b^2*c + 4*a*c^2)*d^2*
x^2 + (b^3 + 4*a*b*c)*d^2*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((2*c*d*x+b*d)**(5/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.40137, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^(3/2),x, algorithm="giac")

[Out]

Done