3.1233 \(\int \frac{(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{12}} \, dx\)

Optimal. Leaf size=118 \[ \frac{16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac{8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (8*(a + b*x + c*x^2)^(7/2))/(99*(b^2 - 4*
a*c)^2*d^12*(b + 2*c*x)^9) + (16*(a + b*x + c*x^2)^(7/2))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^7)

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Rubi [A]  time = 0.056215, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {693, 682} \[ \frac{16 \left (a+b x+c x^2\right )^{7/2}}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^7}+\frac{8 \left (a+b x+c x^2\right )^{7/2}}{99 d^{12} \left (b^2-4 a c\right )^2 (b+2 c x)^9}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (8*(a + b*x + c*x^2)^(7/2))/(99*(b^2 - 4*
a*c)^2*d^12*(b + 2*c*x)^9) + (16*(a + b*x + c*x^2)^(7/2))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^7)

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 )^{5/2}}{(b d+2 c d x)^{12}} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac{4 \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx}{11 \left (b^2-4 a c\right ) d^2}\\ &=\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac{8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac{8 \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx}{99 \left (b^2-4 a c\right )^2 d^4}\\ &=\frac{2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac{8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac{16 \left (a+b x+c x^2\right )^{7/2}}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0815025, size = 110, normalized size = 0.93 \[ \frac{2 (a+x (b+c x))^{7/2} \left (16 c^2 \left (63 a^2-28 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (38 c x^2-77 a\right )+64 b c^2 x \left (4 c x^2-7 a\right )+176 b^3 c x+99 b^4\right )}{693 d^{12} \left (b^2-4 a c\right )^3 (b+2 c x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]

[Out]

(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c*x^2) + 8*b^2*c*(-77*a + 38*c*x^2) +
16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^4)))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)

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Maple [A]  time = 0.048, size = 133, normalized size = 1.1 \begin{align*} -{\frac{256\,{c}^{4}{x}^{4}+512\,b{c}^{3}{x}^{3}-896\,a{c}^{3}{x}^{2}+608\,{b}^{2}{c}^{2}{x}^{2}-896\,ab{c}^{2}x+352\,{b}^{3}cx+2016\,{a}^{2}{c}^{2}-1232\,ac{b}^{2}+198\,{b}^{4}}{693\, \left ( 2\,cx+b \right ) ^{11}{d}^{12} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \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)^12,x)

[Out]

-2/693*(128*c^4*x^4+256*b*c^3*x^3-448*a*c^3*x^2+304*b^2*c^2*x^2-448*a*b*c^2*x+176*b^3*c*x+1008*a^2*c^2-616*a*b
^2*c+99*b^4)*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^11/d^12/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

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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)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^12,x, algorithm="fricas")

[Out]

Timed out

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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)**12,x)

[Out]

Timed out

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Giac [B]  time = 11.9111, size = 3201, normalized size = 27.13 \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)^12,x, algorithm="giac")

[Out]

1/5544*(29568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*c^(17/2) + 236544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*
b*c^8 + 868560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b^2*c^(15/2) + 73920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^14*a*c^(17/2) + 1940400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^3*c^7 + 517440*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^13*a*b*c^8 + 2953104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^4*c^(13/2) + 1600368*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^12*a*b^2*c^(15/2) + 162624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a^2*c^(17/2) + 3256176*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^11*b^5*c^6 + 2875488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^3*c^7 + 975744*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b*c^8 + 2709168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^6*c^(11/2)
+ 3307920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^4*c^(13/2) + 2583504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^10*a^2*b^2*c^(15/2) + 133056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*c^(17/2) + 1755600*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^9*b^7*c^5 + 2513280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^5*c^6 + 3973200*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^9*a^2*b^3*c^7 + 665280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b*c^8 + 910800*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^8*b^8*c^(9/2) + 1227600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^6*c^(11/2) + 394
4160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b^4*c^(13/2) + 1401840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^
3*b^2*c^(15/2) + 95040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*c^(17/2) + 387024*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^7*b^9*c^4 + 319968*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^7*c^5 + 2670624*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*a^2*b^5*c^6 + 1615680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b^3*c^7 + 380160*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^7*a^4*b*c^8 + 136488*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^10*c^(7/2) - 20592*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^6*a*b^8*c^(9/2) + 1284624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^6*c^(11/2) +
1092960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^4*c^(13/2) + 641520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*
a^4*b^2*c^(15/2) + 19008*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^5*c^(17/2) + 39864*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^5*b^11*c^3 - 58080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^9*c^4 + 460944*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*a^2*b^7*c^5 + 418176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^5*c^6 + 594000*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^5*a^4*b^3*c^7 + 57024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^5*b*c^8 + 9460*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*b^12*c^(5/2) - 27720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^10*c^(7/2) + 132000*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^8*c^(9/2) + 64240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^6*c^(1
1/2) + 330000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^4*c^(13/2) + 66000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*a^5*b^2*c^(15/2) + 3520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^6*c^(17/2) + 1760*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*b^13*c^2 - 7920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^11*c^3 + 31680*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^3*a^2*b^9*c^4 - 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^7*c^5 + 113520*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*a^4*b^5*c^6 + 36960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b^3*c^7 + 7040*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a^6*b*c^8 + 242*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^14*c^(3/2) - 1496*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*a*b^12*c^(5/2) + 6072*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^10*c^(7/2) - 8
800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^8*c^(9/2) + 24640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^
6*c^(11/2) + 8976*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^4*c^(13/2) + 6512*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*a^6*b^2*c^(15/2) - 704*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^7*c^(17/2) + 22*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^15*c - 176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^13*c^2 + 792*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^2*b^11*c^3 - 1760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^9*c^4 + 3520*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a^4*b^7*c^5 + 2992*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b^3*c^7 - 704*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^7*b*c^8 + b^16*sqrt(c) - 10*a*b^14*c^(3/2) + 52*a^2*b^12*c^(5/2) - 152*a^3*b^10*c^(7/2) + 320*a^4*b^8*c
^(9/2) - 320*a^5*b^6*c^(11/2) + 640*a^6*b^4*c^(13/2) - 304*a^7*b^2*c^(15/2) + 64*a^8*c^(17/2))/((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)^11*c^4*d^12)