3.1229 \(\int \frac{(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^8} \, dx\)
Optimal. Leaf size=39 \[ \frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
[Out]
(2*(a + b*x + c*x^2)^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)
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Rubi [A] time = 0.0157464, antiderivative size = 39, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used =
{682} \[ \frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]
[Out]
(2*(a + b*x + c*x^2)^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)
Rule 682
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}\\ \end{align*}
Mathematica [A] time = 0.0271806, size = 38, normalized size = 0.97 \[ \frac{2 (a+x (b+c x))^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]
[Out]
(2*(a + x*(b + c*x))^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)
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Maple [A] time = 0.042, size = 38, normalized size = 1. \begin{align*} -{\frac{2}{7\, \left ( 2\,cx+b \right ) ^{7}{d}^{8} \left ( 4\,ac-{b}^{2} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}
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)^8,x)
[Out]
-2/7*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^7/d^8/(4*a*c-b^2)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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)^8,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 112.052, size = 563, normalized size = 14.44 \begin{align*} \frac{2 \,{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt{c x^{2} + b x + a}}{7 \,{\left (128 \,{\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{8} x^{7} + 448 \,{\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{8} x^{6} + 672 \,{\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{8} x^{5} + 560 \,{\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{8} x^{4} + 280 \,{\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{8} x^{3} + 84 \,{\left (b^{7} c^{2} - 4 \, a b^{5} c^{3}\right )} d^{8} x^{2} + 14 \,{\left (b^{8} c - 4 \, a b^{6} c^{2}\right )} d^{8} x +{\left (b^{9} - 4 \, a b^{7} c\right )} d^{8}\right )}} \end{align*}
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)^8,x, algorithm="fricas")
[Out]
2/7*(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)
*x^2)*sqrt(c*x^2 + b*x + a)/(128*(b^2*c^7 - 4*a*c^8)*d^8*x^7 + 448*(b^3*c^6 - 4*a*b*c^7)*d^8*x^6 + 672*(b^4*c^
5 - 4*a*b^2*c^6)*d^8*x^5 + 560*(b^5*c^4 - 4*a*b^3*c^5)*d^8*x^4 + 280*(b^6*c^3 - 4*a*b^4*c^4)*d^8*x^3 + 84*(b^7
*c^2 - 4*a*b^5*c^3)*d^8*x^2 + 14*(b^8*c - 4*a*b^6*c^2)*d^8*x + (b^9 - 4*a*b^7*c)*d^8)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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)**8,x)
[Out]
Timed out
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Giac [B] time = 2.51329, size = 1683, normalized size = 43.15 \begin{align*} \text{result too large to display} \end{align*}
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)^8,x, algorithm="giac")
[Out]
1/448*(448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*c^(13/2) + 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b*c^6
+ 7392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^2*c^(11/2) + 12320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*
c^5 + 14000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(9/2) - 1120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b
^2*c^(11/2) + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(13/2) + 11648*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^7*b^5*c^4 - 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^5 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^7*a^2*b*c^6 + 7448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*c^(7/2) - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^6*a*b^4*c^(9/2) + 15680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^2*c^(11/2) + 3752*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*b^7*c^3 - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^5*c^4 + 15680*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a^2*b^3*c^5 + 1484*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^8*c^(5/2) - 4984*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*a*b^6*c^(7/2) + 10304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^(9/2) - 1344*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^2*c^(11/2) + 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^(13/2) + 4
48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^9*c^2 - 2128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^7*c^3 + 4928
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c^4 - 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^3*c^5 +
2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^6 + 98*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^10*c^(3/2) -
616*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^8*c^(5/2) + 1736*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^6*
c^(7/2) - 2016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^4*c^(9/2) + 2016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a^4*b^2*c^(11/2) + 14*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^11*c - 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*a*b^9*c^2 + 392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^7*c^3 - 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3
*b^5*c^4 + 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3*c^5 + b^12*sqrt(c) - 10*a*b^10*c^(3/2) + 44*a^2*b^8
*c^(5/2) - 104*a^3*b^6*c^(7/2) + 144*a^4*b^4*c^(9/2) - 96*a^5*b^2*c^(11/2) + 64*a^6*c^(13/2))/((2*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^7*c^4*d^8)