### 3.1214 $$\int \frac{(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^8} \, dx$$

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 = 0.0348012, 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, Rules used = {693, 682} $\frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7}$

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)

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

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 )^{3/2}}{(b d+2 c d x)^8} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac{2 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx}{7 \left (b^2-4 a c\right ) d^2}\\ &=\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0367531, 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 = 0.044, size = 70, normalized size = 0.9 \begin{align*} -{\frac{-16\,{c}^{2}{x}^{2}-16\,bcx+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}}}} \end{align*}

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(-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)^(3/2)/(2*c*d*x+b*d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 119.981, size = 841, normalized size = 10.65 \begin{align*} \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 )}} \end{align*}

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. \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)**(3/2)/(2*c*d*x+b*d)**8,x)

[Out]

Timed out

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Giac [B]  time = 2.00361, size = 1354, normalized size = 17.14 \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)^(3/2)/(2*c*d*x+b*d)^8,x, algorithm="giac")

[Out]

1/280*(560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*c^(11/2) + 2800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b*c^5
+ 6160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^(9/2) + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(11/2
) + 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^4 + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^5 +
6440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(7/2) + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(9
/2) + 1120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(11/2) + 3640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5
*c^3 + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^4 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b
*c^5 + 1484*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^(5/2) + 392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^
4*c^(7/2) + 4032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^(9/2) + 224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*a^3*c^(11/2) + 448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^7*c^2 - 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^3*a*b^5*c^3 + 2464*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^3*c^4 + 448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*a^3*b*c^5 + 98*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*c^(3/2) - 224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a*b^6*c^(5/2) + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c^(7/2) + 224*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a^3*b^2*c^(9/2) + 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*c^(11/2) + 14*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*b^9*c - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7*c^2 + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^2*b^5*c^3 + 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^5 + b^10*sqrt(c) - 6*a*b^8*c^(3/2) + 20*a^2
*b^6*c^(5/2) - 24*a^3*b^4*c^(7/2) + 48*a^4*b^2*c^(9/2) - 16*a^5*c^(11/2))/((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^3*d^8)