3.1203 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

[Out]

(2*(a + b*x + c*x^2)^(5/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7) + (4*(a + b*x +
c*x^2)^(5/2))/(35*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^5)

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Rubi [A]  time = 0.110449, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + b*x + c*x^2)^(5/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7) + (4*(a + b*x +
c*x^2)^(5/2))/(35*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^5)

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Rubi in Sympy [A]  time = 27.3768, size = 75, normalized size = 0.95 \[ \frac{4 \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{35 d^{8} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right )^{2}} + \frac{2 \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7 d^{8} \left (b + 2 c x\right )^{7} \left (- 4 a c + b^{2}\right )} \]

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)**8,x)

[Out]

4*(a + b*x + c*x**2)**(5/2)/(35*d**8*(b + 2*c*x)**5*(-4*a*c + b**2)**2) + 2*(a +
b*x + c*x**2)**(5/2)/(7*d**8*(b + 2*c*x)**7*(-4*a*c + b**2))

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Mathematica [A]  time = 0.154206, size = 62, normalized size = 0.78 \[ \frac{2 (a+x (b+c x))^{5/2} \left (4 c \left (2 c x^2-5 a\right )+7 b^2+8 b c x\right )}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + x*(b + c*x))^(5/2)*(7*b^2 + 8*b*c*x + 4*c*(-5*a + 2*c*x^2)))/(35*(b^2 -
4*a*c)^2*d^8*(b + 2*c*x)^7)

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Maple [A]  time = 0.009, size = 70, normalized size = 0.9 \[ -{\frac{-16\,{c}^{2}{x}^{2}-16\,bxc+40\,ac-14\,{b}^{2}}{35\, \left ( 2\,cx+b \right ) ^{7}{d}^{8} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \]

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)^8,x)

[Out]

-2/35*(-8*c^2*x^2-8*b*c*x+20*a*c-7*b^2)*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^7/d^8/(16*
a^2*c^2-8*a*b^2*c+b^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.14553, size = 537, normalized size = 6.8 \[ \frac{2 \,{\left (8 \, c^{4} x^{6} + 24 \, b c^{3} x^{5} +{\left (31 \, b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 7 \, a^{2} b^{2} - 20 \, a^{3} c + 2 \,{\left (11 \, b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (7 \, b^{4} + 10 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (7 \, a b^{3} - 16 \, a^{2} b c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{35 \,{\left (128 \,{\left (b^{4} c^{7} - 8 \, a b^{2} c^{8} + 16 \, a^{2} c^{9}\right )} d^{8} x^{7} + 448 \,{\left (b^{5} c^{6} - 8 \, a b^{3} c^{7} + 16 \, a^{2} b c^{8}\right )} d^{8} x^{6} + 672 \,{\left (b^{6} c^{5} - 8 \, a b^{4} c^{6} + 16 \, a^{2} b^{2} c^{7}\right )} d^{8} x^{5} + 560 \,{\left (b^{7} c^{4} - 8 \, a b^{5} c^{5} + 16 \, a^{2} b^{3} c^{6}\right )} d^{8} x^{4} + 280 \,{\left (b^{8} c^{3} - 8 \, a b^{6} c^{4} + 16 \, a^{2} b^{4} c^{5}\right )} d^{8} x^{3} + 84 \,{\left (b^{9} c^{2} - 8 \, a b^{7} c^{3} + 16 \, a^{2} b^{5} c^{4}\right )} d^{8} x^{2} + 14 \,{\left (b^{10} c - 8 \, a b^{8} c^{2} + 16 \, a^{2} b^{6} c^{3}\right )} d^{8} x +{\left (b^{11} - 8 \, a b^{9} c + 16 \, a^{2} b^{7} c^{2}\right )} d^{8}\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

2/35*(8*c^4*x^6 + 24*b*c^3*x^5 + (31*b^2*c^2 - 4*a*c^3)*x^4 + 7*a^2*b^2 - 20*a^3
*c + 2*(11*b^3*c - 4*a*b*c^2)*x^3 + (7*b^4 + 10*a*b^2*c - 32*a^2*c^2)*x^2 + 2*(7
*a*b^3 - 16*a^2*b*c)*x)*sqrt(c*x^2 + b*x + a)/(128*(b^4*c^7 - 8*a*b^2*c^8 + 16*a
^2*c^9)*d^8*x^7 + 448*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^8*x^6 + 672*(b^6*
c^5 - 8*a*b^4*c^6 + 16*a^2*b^2*c^7)*d^8*x^5 + 560*(b^7*c^4 - 8*a*b^5*c^5 + 16*a^
2*b^3*c^6)*d^8*x^4 + 280*(b^8*c^3 - 8*a*b^6*c^4 + 16*a^2*b^4*c^5)*d^8*x^3 + 84*(
b^9*c^2 - 8*a*b^7*c^3 + 16*a^2*b^5*c^4)*d^8*x^2 + 14*(b^10*c - 8*a*b^8*c^2 + 16*
a^2*b^6*c^3)*d^8*x + (b^11 - 8*a*b^9*c + 16*a^2*b^7*c^2)*d^8)

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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)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.893775, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

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)^8,x, algorithm="giac")

[Out]

sage0*x