3.1282 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac {3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{64 c^4 d^5}-\frac {\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}+\frac {\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac {(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]

[Out]

1/448*(-4*a*c+b^2)^3/c^4/d/(2*c*d*x+b*d)^(7/2)-1/64*(-4*a*c+b^2)^2/c^4/d^3/(2*c*d*x+b*d)^(3/2)+1/320*(2*c*d*x+
b*d)^(5/2)/c^4/d^7-3/64*(-4*a*c+b^2)*(2*c*d*x+b*d)^(1/2)/c^4/d^5

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Rubi [A]  time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {683} \[ -\frac {\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{64 c^4 d^5}+\frac {\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac {(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(b^2 - 4*a*c)^3/(448*c^4*d*(b*d + 2*c*d*x)^(7/2)) - (b^2 - 4*a*c)^2/(64*c^4*d^3*(b*d + 2*c*d*x)^(3/2)) - (3*(b
^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x])/(64*c^4*d^5) + (b*d + 2*c*d*x)^(5/2)/(320*c^4*d^7)

Rule 683

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{9/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{5/2}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^4 \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{64 c^3 d^6}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}-\frac {\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{64 c^4 d^5}+\frac {(b d+2 c d x)^{5/2}}{320 c^4 d^7}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 83, normalized size = 0.69 \[ \frac {-105 \left (b^2-4 a c\right ) (b+2 c x)^4-35 \left (b^2-4 a c\right )^2 (b+2 c x)^2+5 \left (b^2-4 a c\right )^3+7 (b+2 c x)^6}{2240 c^4 d (d (b+2 c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(5*(b^2 - 4*a*c)^3 - 35*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 105*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 7*(b + 2*c*x)^6)/(22
40*c^4*d*(d*(b + 2*c*x))^(7/2))

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fricas [A]  time = 0.84, size = 210, normalized size = 1.74 \[ \frac {{\left (7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 105 \, a c^{5} x^{4} - 2 \, b^{6} + 10 \, a b^{4} c - 5 \, a^{2} b^{2} c^{2} - 5 \, a^{3} c^{3} - 35 \, {\left (b^{3} c^{3} - 6 \, a b c^{4}\right )} x^{3} - 35 \, {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} - 7 \, {\left (2 \, b^{5} c - 10 \, a b^{3} c^{2} + 5 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{35 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(9/2),x, algorithm="fricas")

[Out]

1/35*(7*c^6*x^6 + 21*b*c^5*x^5 + 105*a*c^5*x^4 - 2*b^6 + 10*a*b^4*c - 5*a^2*b^2*c^2 - 5*a^3*c^3 - 35*(b^3*c^3
- 6*a*b*c^4)*x^3 - 35*(b^4*c^2 - 5*a*b^2*c^3 + a^2*c^4)*x^2 - 7*(2*b^5*c - 10*a*b^3*c^2 + 5*a^2*b*c^3)*x)*sqrt
(2*c*d*x + b*d)/(16*c^8*d^5*x^4 + 32*b*c^7*d^5*x^3 + 24*b^2*c^6*d^5*x^2 + 8*b^3*c^5*d^5*x + b^4*c^4*d^5)

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giac [A]  time = 0.69, size = 186, normalized size = 1.54 \[ \frac {b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2} - 7 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} + 56 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 112 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{448 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{4} d^{3}} - \frac {15 \, \sqrt {2 \, c d x + b d} b^{2} c^{16} d^{30} - 60 \, \sqrt {2 \, c d x + b d} a c^{17} d^{30} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{16} d^{28}}{320 \, c^{20} d^{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(9/2),x, algorithm="giac")

[Out]

1/448*(b^6*d^2 - 12*a*b^4*c*d^2 + 48*a^2*b^2*c^2*d^2 - 64*a^3*c^3*d^2 - 7*(2*c*d*x + b*d)^2*b^4 + 56*(2*c*d*x
+ b*d)^2*a*b^2*c - 112*(2*c*d*x + b*d)^2*a^2*c^2)/((2*c*d*x + b*d)^(7/2)*c^4*d^3) - 1/320*(15*sqrt(2*c*d*x + b
*d)*b^2*c^16*d^30 - 60*sqrt(2*c*d*x + b*d)*a*c^17*d^30 - (2*c*d*x + b*d)^(5/2)*c^16*d^28)/(c^20*d^35)

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maple [A]  time = 0.05, size = 163, normalized size = 1.35 \[ -\frac {\left (2 c x +b \right ) \left (-7 c^{6} x^{6}-21 b \,c^{5} x^{5}-105 a \,c^{5} x^{4}-210 a b \,c^{4} x^{3}+35 b^{3} c^{3} x^{3}+35 a^{2} c^{4} x^{2}-175 a \,b^{2} c^{3} x^{2}+35 b^{4} c^{2} x^{2}+35 a^{2} b \,c^{3} x -70 a \,b^{3} c^{2} x +14 b^{5} c x +5 a^{3} c^{3}+5 a^{2} b^{2} c^{2}-10 a \,b^{4} c +2 b^{6}\right )}{35 \left (2 c d x +b d \right )^{\frac {9}{2}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(9/2),x)

[Out]

-1/35*(2*c*x+b)*(-7*c^6*x^6-21*b*c^5*x^5-105*a*c^5*x^4-210*a*b*c^4*x^3+35*b^3*c^3*x^3+35*a^2*c^4*x^2-175*a*b^2
*c^3*x^2+35*b^4*c^2*x^2+35*a^2*b*c^3*x-70*a*b^3*c^2*x+14*b^5*c*x+5*a^3*c^3+5*a^2*b^2*c^2-10*a*b^4*c+2*b^6)/c^4
/(2*c*d*x+b*d)^(9/2)

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maxima [A]  time = 1.43, size = 142, normalized size = 1.17 \[ -\frac {\frac {5 \, {\left (7 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{3} d^{2}} + \frac {7 \, {\left (15 \, \sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )}}{c^{3} d^{6}}}{2240 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(9/2),x, algorithm="maxima")

[Out]

-1/2240*(5*(7*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^2 - (b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c
^3)*d^2)/((2*c*d*x + b*d)^(7/2)*c^3*d^2) + 7*(15*sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^2 - (2*c*d*x + b*d)^(5/2)
)/(c^3*d^6))/(c*d)

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mupad [B]  time = 0.51, size = 159, normalized size = 1.31 \[ -\frac {5\,a^3\,c^3+5\,a^2\,b^2\,c^2+35\,a^2\,b\,c^3\,x+35\,a^2\,c^4\,x^2-10\,a\,b^4\,c-70\,a\,b^3\,c^2\,x-175\,a\,b^2\,c^3\,x^2-210\,a\,b\,c^4\,x^3-105\,a\,c^5\,x^4+2\,b^6+14\,b^5\,c\,x+35\,b^4\,c^2\,x^2+35\,b^3\,c^3\,x^3-21\,b\,c^5\,x^5-7\,c^6\,x^6}{35\,c^4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(9/2),x)

[Out]

-(2*b^6 + 5*a^3*c^3 - 7*c^6*x^6 - 105*a*c^5*x^4 - 21*b*c^5*x^5 + 5*a^2*b^2*c^2 + 35*a^2*c^4*x^2 + 35*b^4*c^2*x
^2 + 35*b^3*c^3*x^3 - 10*a*b^4*c + 14*b^5*c*x - 175*a*b^2*c^3*x^2 - 70*a*b^3*c^2*x + 35*a^2*b*c^3*x - 210*a*b*
c^4*x^3)/(35*c^4*d*(b*d + 2*c*d*x)^(7/2))

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sympy [A]  time = 11.27, size = 1394, normalized size = 11.52 \[ \begin {cases} - \frac {5 a^{3} c^{3} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {5 a^{2} b^{2} c^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 a^{2} b c^{3} x \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 a^{2} c^{4} x^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {10 a b^{4} c \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {70 a b^{3} c^{2} x \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {175 a b^{2} c^{3} x^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {210 a b c^{4} x^{3} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {105 a c^{5} x^{4} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {2 b^{6} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {14 b^{5} c x \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 b^{4} c^{2} x^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 b^{3} c^{3} x^{3} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {21 b c^{5} x^{5} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {7 c^{6} x^{6} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} & \text {for}\: c \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}}{\left (b d\right )^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(9/2),x)

[Out]

Piecewise((-5*a**3*c**3*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**
2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 5*a**2*b**2*c**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 28
0*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 35*a**2*b*c**3*x*
sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**
3 + 560*c**8*d**5*x**4) - 35*a**2*c**4*x**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 84
0*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 10*a*b**4*c*sqrt(b*d + 2*c*d*x)/(35*b**4
*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 70
*a*b**3*c**2*x*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*
b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 175*a*b**2*c**3*x**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**
3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 210*a*b*c**4*x**3*sqrt
(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 +
560*c**8*d**5*x**4) + 105*a*c**5*x**4*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2
*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 2*b**6*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5
+ 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 14*b**5*c*x*s
qrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3
+ 560*c**8*d**5*x**4) - 35*b**4*c**2*x**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840
*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 35*b**3*c**3*x**3*sqrt(b*d + 2*c*d*x)/(35
*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4)
+ 21*b*c**5*x**5*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 11
20*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 7*c**6*x**6*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5
*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4), Ne(c, 0)), ((a**3*x + 3*a**2*
b*x**2/2 + a*b**2*x**3 + b**3*x**4/4)/(b*d)**(9/2), True))

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