3.1344 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{17/2}} \, dx\)

Optimal. Leaf size=258 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{924 c^4 d^{17/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{462 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{308 c^3 d^5 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{66 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{15 c d (b d+2 c d x)^{15/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(308*c^3*d^5*(b*d + 2*c*d*x)^(7/2)) + Sqrt[a + b*x + c*x^
2]/(462*c^3*(b^2 - 4*a*c)*d^7*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2)/(
66*c^2*d^3*(b*d + 2*c*d*x)^(11/2)) - (a + b*x + c*x^2)^(5/2)/(15*c*d*(b*d + 2*c*
d*x)^(15/2)) + (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sq
rt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(924*c^4*(b^2 - 4*a*c)^(3
/4)*d^(17/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.614478, antiderivative size = 258, 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 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{924 c^4 d^{17/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{462 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{308 c^3 d^5 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{66 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{15 c d (b d+2 c d x)^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(17/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(308*c^3*d^5*(b*d + 2*c*d*x)^(7/2)) + Sqrt[a + b*x + c*x^
2]/(462*c^3*(b^2 - 4*a*c)*d^7*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2)/(
66*c^2*d^3*(b*d + 2*c*d*x)^(11/2)) - (a + b*x + c*x^2)^(5/2)/(15*c*d*(b*d + 2*c*
d*x)^(15/2)) + (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sq
rt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(924*c^4*(b^2 - 4*a*c)^(3
/4)*d^(17/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 141.825, size = 241, normalized size = 0.93 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{15 c d \left (b d + 2 c d x\right )^{\frac{15}{2}}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{66 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{11}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{308 c^{3} d^{5} \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{\sqrt{a + b x + c x^{2}}}{462 c^{3} d^{7} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{924 c^{4} d^{\frac{17}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(17/2),x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(15*c*d*(b*d + 2*c*d*x)**(15/2)) - (a + b*x + c*x**2)
**(3/2)/(66*c**2*d**3*(b*d + 2*c*d*x)**(11/2)) - sqrt(a + b*x + c*x**2)/(308*c**
3*d**5*(b*d + 2*c*d*x)**(7/2)) + sqrt(a + b*x + c*x**2)/(462*c**3*d**7*(-4*a*c +
b**2)*(b*d + 2*c*d*x)**(3/2)) + sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*ellip
tic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(924*c**4*
d**(17/2)*(-4*a*c + b**2)**(3/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.32658, size = 213, normalized size = 0.83 \[ \frac{-c (b+2 c x) (a+x (b+c x)) \left (207 \left (b^2-4 a c\right ) (b+2 c x)^4-224 \left (b^2-4 a c\right )^2 (b+2 c x)^2+77 \left (b^2-4 a c\right )^3-40 (b+2 c x)^6\right )+\frac{20 i (b+2 c x)^{19/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{18480 c^4 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} (d (b+2 c x))^{17/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(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 - 224*(b^2 - 4*a*c)^2*(b
+ 2*c*x)^2 + 207*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 40*(b + 2*c*x)^6)) + ((20*I)*(b +
2*c*x)^(19/2)*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sqr
t[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/Sqrt[-Sqrt[b^2 - 4*a*c]])/(18480*c^
4*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(17/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.034, size = 1431, normalized size = 5.6 \[ \text{result too large to display} \]

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)^(17/2),x)

[Out]

-1/9240*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(7904*x^6*a*c^7+13792*x^4*a^2*c^
6+9632*x^2*a^3*c^5+3032*x^5*b^3*c^5-150*x^2*b^6*c^2-948*x^4*b^4*c^4-10*x*b^7*c+5
120*x^7*b*c^7+6984*x^6*b^2*c^6-976*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-56*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)*Ell
ipticF(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+2240
*(-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+2464*a^4*c^4+23712*x^5*a*b*c^6+22744*x^4*a*b^2*c^5+2
7584*x^3*a^2*b*c^5+5968*x^3*a*b^3*c^4+13464*x^2*a^2*b^2*c^4-1128*x^2*a*b^4*c^3+9
632*x*a^3*b*c^4-328*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/(4*a*c-b^2)/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{17}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(17/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(17/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(17/2),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)/((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)**(5/2)/(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. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{17}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(17/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(17/2), x)