3.1339 \(\int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=274 \[ -\frac{d^{3/2} \left (b^2-4 a c\right )^{17/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{924 c^4 \sqrt{a+b x+c x^2}}-\frac{d \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{462 c^3}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{308 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{66 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{5/2}}{15 c d} \]

[Out]

-((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(462*c^3) + ((b^2
- 4*a*c)^2*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(308*c^3*d) - ((b^2 - 4
*a*c)*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(66*c^2*d) + ((b*d + 2*c*d*
x)^(5/2)*(a + b*x + c*x^2)^(5/2))/(15*c*d) - ((b^2 - 4*a*c)^(17/4)*d^(3/2)*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])/(924*c^4*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.65868, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{d^{3/2} \left (b^2-4 a c\right )^{17/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{924 c^4 \sqrt{a+b x+c x^2}}-\frac{d \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{462 c^3}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{308 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{66 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{5/2}}{15 c d} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(462*c^3) + ((b^2
- 4*a*c)^2*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(308*c^3*d) - ((b^2 - 4
*a*c)*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(66*c^2*d) + ((b*d + 2*c*d*
x)^(5/2)*(a + b*x + c*x^2)^(5/2))/(15*c*d) - ((b^2 - 4*a*c)^(17/4)*d^(3/2)*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])/(924*c^4*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 131.903, size = 257, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{15 c d} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{66 c^{2} d} - \frac{d \left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{462 c^{3}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{308 c^{3} d} - \frac{d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{17}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{924 c^{4} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*d + 2*c*d*x)**(5/2)*(a + b*x + c*x**2)**(5/2)/(15*c*d) - (-4*a*c + b**2)*(b*d
+ 2*c*d*x)**(5/2)*(a + b*x + c*x**2)**(3/2)/(66*c**2*d) - d*(-4*a*c + b**2)**3*
sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(462*c**3) + (-4*a*c + b**2)**2*(b*d
+ 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/(308*c**3*d) - d**(3/2)*sqrt(c*(a + b*x
+ c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(17/4)*elliptic_f(asin(sqrt(b*d + 2*
c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(924*c**4*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.33421, size = 295, normalized size = 1.08 \[ \frac{(d (b+2 c x))^{3/2} \left (\frac{c (a+x (b+c x)) \left (4 b^2 c^2 \left (87 a^2+930 a c x^2+931 c^2 x^4\right )+16 b c^3 x \left (207 a^2+448 a c x^2+231 c^2 x^4\right )+16 c^3 \left (40 a^3+207 a^2 c x^2+224 a c^2 x^4+77 c^3 x^6\right )+2 b^4 c \left (9 c x^2-35 a\right )+8 b^3 c^2 x \left (17 a+161 c x^2\right )+5 b^6-10 b^5 c x\right )}{b+2 c x}-\frac{5 i \left (b^2-4 a c\right )^4 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} \sqrt{b+2 c x}}\right )}{4620 c^4 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(3/2)*((c*(a + x*(b + c*x))*(5*b^6 - 10*b^5*c*x + 2*b^4*c*(-35*
a + 9*c*x^2) + 8*b^3*c^2*x*(17*a + 161*c*x^2) + 16*b*c^3*x*(207*a^2 + 448*a*c*x^
2 + 231*c^2*x^4) + 4*b^2*c^2*(87*a^2 + 930*a*c*x^2 + 931*c^2*x^4) + 16*c^3*(40*a
^3 + 207*a^2*c*x^2 + 224*a*c^2*x^4 + 77*c^3*x^6)))/(b + 2*c*x) - ((5*I)*(b^2 - 4
*a*c)^4*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sqrt[-Sqrt
[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/(Sqrt[-Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c*x])
))/(4620*c^4*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.036, size = 1055, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9240*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d*(-1280*a^4*b*c^4-696*a^3*b^3*c
^3+140*a^2*b^5*c^2-10*a*b^7*c+5*((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))*(-4*a*c+b^2)^(1/2)*b^8+1280*((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))*(-4*a*c+b^2)^(1/2)*a^4*c
^4-67424*x^6*a*b*c^7-87344*x^5*a*b^2*c^6-68960*x^4*a^2*b*c^6-49800*x^4*a*b^3*c^5
-57104*x^3*a^2*b^2*c^5-10624*x^3*a*b^4*c^4-23712*x^2*a^3*b*c^5-16696*x^2*a^2*b^3
*c^4+152*x^2*a*b^5*c^3-9296*x*a^3*b^2*c^4-688*x*a^2*b^4*c^3+140*x*a*b^6*c^2-4928
*x^9*c^9-80*((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))*(-4*a*c+b^2)^(1/2)*a*b^6*c-22176*x^8*b*c^8-1280*((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))*(-4*a*c+b^2)^(1/2)*a^3*b
^2*c^3-19264*x^7*a*c^8-39536*x^7*b^2*c^7-34888*x^6*b^3*c^6-27584*x^5*a^2*c^7-152
48*x^5*b^4*c^5-2644*x^4*b^5*c^4-15808*x^3*a^3*c^6+4*x^3*b^6*c^3-10*x^2*b^7*c^2-2
560*x*a^4*c^5-10*x*b^8*c+480*((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))*(-4*a*c+b^2)^(1/2)*a^2*b^4*c^2)/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. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (2 \, c^{3} d x^{5} + 5 \, b c^{2} d x^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d x^{3} + a^{2} b d +{\left (b^{3} + 6 \, a b c\right )} d x^{2} + 2 \,{\left (a b^{2} + a^{2} c\right )} d x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((2*c^3*d*x^5 + 5*b*c^2*d*x^4 + 4*(b^2*c + a*c^2)*d*x^3 + a^2*b*d + (b^3
+ 6*a*b*c)*d*x^2 + 2*(a*b^2 + a^2*c)*d*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)**(3/2)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.81422, 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)^(3/2)*(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

Done