3.1332 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{17/2}} \, dx\)

Optimal. Leaf size=268 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(110*c^2*d^3*(b*d + 2*c*d*x)^(11/2)) + Sqrt[a + b*x + c*x
^2]/(385*c^2*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(7/2)) + Sqrt[a + b*x + c*x^2]/(2
31*c^2*(b^2 - 4*a*c)^2*d^7*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2)/(15*
c*d*(b*d + 2*c*d*x)^(15/2)) + (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elli
pticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(462*c^3*(
b^2 - 4*a*c)^(7/4)*d^(17/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.63974, antiderivative size = 268, 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 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(17/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(110*c^2*d^3*(b*d + 2*c*d*x)^(11/2)) + Sqrt[a + b*x + c*x
^2]/(385*c^2*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(7/2)) + Sqrt[a + b*x + c*x^2]/(2
31*c^2*(b^2 - 4*a*c)^2*d^7*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2)/(15*
c*d*(b*d + 2*c*d*x)^(15/2)) + (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elli
pticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(462*c^3*(
b^2 - 4*a*c)^(7/4)*d^(17/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 137.613, size = 252, normalized size = 0.94 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{15 c d \left (b d + 2 c d x\right )^{\frac{15}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{110 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{11}{2}}} + \frac{\sqrt{a + b x + c x^{2}}}{385 c^{2} d^{5} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{\sqrt{a + b x + c x^{2}}}{231 c^{2} d^{7} \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{462 c^{3} d^{\frac{17}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**(17/2),x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(15*c*d*(b*d + 2*c*d*x)**(15/2)) - sqrt(a + b*x + c*x
**2)/(110*c**2*d**3*(b*d + 2*c*d*x)**(11/2)) + sqrt(a + b*x + c*x**2)/(385*c**2*
d**5*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(7/2)) + sqrt(a + b*x + c*x**2)/(231*c**2*
d**7*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(3/2)) + 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)/(462*c**3*d**(17/2)*(-4*a*c + b**2)**(7/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.34211, size = 212, normalized size = 0.79 \[ \frac{c (b+2 c x) (a+x (b+c x)) \left (12 \left (b^2-4 a c\right ) (b+2 c x)^4-119 \left (b^2-4 a c\right )^2 (b+2 c x)^2+77 \left (b^2-4 a c\right )^3+20 (b+2 c x)^6\right )+\frac{10 i (b+2 c x)^{19/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{4620 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{17/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(17/2),x]

[Out]

(c*(b + 2*c*x)*(a + x*(b + c*x))*(77*(b^2 - 4*a*c)^3 - 119*(b^2 - 4*a*c)^2*(b +
2*c*x)^2 + 12*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 20*(b + 2*c*x)^6) + ((10*I)*(b + 2*c
*x)^(19/2)*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sqrt[-S
qrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/Sqrt[-Sqrt[b^2 - 4*a*c]])/(4620*c^3*(b^
2 - 4*a*c)^2*(d*(b + 2*c*x))^(17/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.078, size = 1431, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^(17/2),x)

[Out]

1/4620*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(512*x^6*a*c^7-8384*x^4*a^2*c^6-1
2544*x^2*a^3*c^5+8576*x^5*b^3*c^5-150*x^2*b^6*c^2+4596*x^4*b^4*c^4-10*x*b^7*c+51
20*x^7*b*c^7+8832*x^6*b^2*c^6+872*x^3*b^5*c^3+640*(-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))*x^7*c^7+1280*
x^8*c^8+1400*(-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))*x^3*b^4*c^3+420*(-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))*x^2*b^5*c^2+70*(-
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))*x*b^6*c+1792*a^3*b^2*c^3-20*a^2*b^4*c^2-10*a*b^6*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)*El
lipticF(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^7+2800*(-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))*x^4*b^3*c^4+3360*(-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))*x^5*b^2*c^5+224
0*(-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))*x^6*b*c^6-4928*a^4*c^4+1536*x^5*a*b*c^6+6112*x^4*a*b^2*c^5-16
768*x^3*a^2*b*c^5+9664*x^3*a*b^3*c^4-3168*x^2*a^2*b^2*c^4+4416*x^2*a*b^4*c^3-125
44*x*a^3*b*c^4+5216*x*a^2*b^3*c^3-160*x*a*b^5*c^2)/d^9/(2*c^2*x^3+3*b*c*x^2+2*a*
c*x+b^2*x+a*b)/(2*c*x+b)^7/c^3/(4*a*c-b^2)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{17}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(17/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(17/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (256 \, c^{8} d^{8} x^{8} + 1024 \, b c^{7} d^{8} x^{7} + 1792 \, b^{2} c^{6} d^{8} x^{6} + 1792 \, b^{3} c^{5} d^{8} x^{5} + 1120 \, b^{4} c^{4} d^{8} x^{4} + 448 \, b^{5} c^{3} d^{8} x^{3} + 112 \, b^{6} c^{2} d^{8} x^{2} + 16 \, b^{7} c d^{8} x + b^{8} d^{8}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(17/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/2)/((256*c^8*d^8*x^8 + 1024*b*c^7*d^8*x^7 + 1792*b
^2*c^6*d^8*x^6 + 1792*b^3*c^5*d^8*x^5 + 1120*b^4*c^4*d^8*x^4 + 448*b^5*c^3*d^8*x
^3 + 112*b^6*c^2*d^8*x^2 + 16*b^7*c*d^8*x + b^8*d^8)*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)**(3/2)/(2*c*d*x+b*d)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{17}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(17/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(17/2), x)