3.1276 \(\int \frac {(a+b x+c x^2)^2}{(b d+2 c d x)^{11/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac {b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac {1}{16 c^3 d^5 \sqrt {b d+2 c d x}} \]

[Out]

-1/144*(-4*a*c+b^2)^2/c^3/d/(2*c*d*x+b*d)^(9/2)+1/40*(-4*a*c+b^2)/c^3/d^3/(2*c*d*x+b*d)^(5/2)-1/16/c^3/d^5/(2*
c*d*x+b*d)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 88, 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} \[ \frac {b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac {\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac {1}{16 c^3 d^5 \sqrt {b d+2 c d x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(11/2),x]

[Out]

-(b^2 - 4*a*c)^2/(144*c^3*d*(b*d + 2*c*d*x)^(9/2)) + (b^2 - 4*a*c)/(40*c^3*d^3*(b*d + 2*c*d*x)^(5/2)) - 1/(16*
c^3*d^5*Sqrt[b*d + 2*c*d*x])

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 )^2}{(b d+2 c d x)^{11/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{11/2}}+\frac {-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{7/2}}+\frac {1}{16 c^2 d^4 (b d+2 c d x)^{3/2}}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}+\frac {b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac {1}{16 c^3 d^5 \sqrt {b d+2 c d x}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 63, normalized size = 0.72 \[ \frac {18 \left (b^2-4 a c\right ) (b+2 c x)^2-5 \left (b^2-4 a c\right )^2-45 (b+2 c x)^4}{720 c^3 d (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(11/2),x]

[Out]

(-5*(b^2 - 4*a*c)^2 + 18*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 45*(b + 2*c*x)^4)/(720*c^3*d*(d*(b + 2*c*x))^(9/2))

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fricas [B]  time = 1.03, size = 162, normalized size = 1.84 \[ -\frac {{\left (45 \, c^{4} x^{4} + 90 \, b c^{3} x^{3} + 2 \, b^{4} + 2 \, a b^{2} c + 5 \, a^{2} c^{2} + 9 \, {\left (7 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2} + 18 \, {\left (b^{3} c + a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{45 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(11/2),x, algorithm="fricas")

[Out]

-1/45*(45*c^4*x^4 + 90*b*c^3*x^3 + 2*b^4 + 2*a*b^2*c + 5*a^2*c^2 + 9*(7*b^2*c^2 + 2*a*c^3)*x^2 + 18*(b^3*c + a
*b*c^2)*x)*sqrt(2*c*d*x + b*d)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 +
10*b^4*c^4*d^6*x + b^5*c^3*d^6)

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giac [A]  time = 0.21, size = 99, normalized size = 1.12 \[ -\frac {5 \, b^{4} d^{4} - 40 \, a b^{2} c d^{4} + 80 \, a^{2} c^{2} d^{4} - 18 \, {\left (2 \, c d x + b d\right )}^{2} b^{2} d^{2} + 72 \, {\left (2 \, c d x + b d\right )}^{2} a c d^{2} + 45 \, {\left (2 \, c d x + b d\right )}^{4}}{720 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(11/2),x, algorithm="giac")

[Out]

-1/720*(5*b^4*d^4 - 40*a*b^2*c*d^4 + 80*a^2*c^2*d^4 - 18*(2*c*d*x + b*d)^2*b^2*d^2 + 72*(2*c*d*x + b*d)^2*a*c*
d^2 + 45*(2*c*d*x + b*d)^4)/((2*c*d*x + b*d)^(9/2)*c^3*d^5)

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maple [A]  time = 0.04, size = 96, normalized size = 1.09 \[ -\frac {\left (2 c x +b \right ) \left (45 c^{4} x^{4}+90 b \,c^{3} x^{3}+18 a \,c^{3} x^{2}+63 x^{2} b^{2} c^{2}+18 a b \,c^{2} x +18 x \,b^{3} c +5 a^{2} c^{2}+2 a \,b^{2} c +2 b^{4}\right )}{45 \left (2 c d x +b d \right )^{\frac {11}{2}} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(11/2),x)

[Out]

-1/45*(2*c*x+b)*(45*c^4*x^4+90*b*c^3*x^3+18*a*c^3*x^2+63*b^2*c^2*x^2+18*a*b*c^2*x+18*b^3*c*x+5*a^2*c^2+2*a*b^2
*c+2*b^4)/c^3/(2*c*d*x+b*d)^(11/2)

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maxima [A]  time = 1.32, size = 81, normalized size = 0.92 \[ \frac {18 \, {\left (2 \, c d x + b d\right )}^{2} {\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{4} - 45 \, {\left (2 \, c d x + b d\right )}^{4}}{720 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(11/2),x, algorithm="maxima")

[Out]

1/720*(18*(2*c*d*x + b*d)^2*(b^2 - 4*a*c)*d^2 - 5*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^4 - 45*(2*c*d*x + b*d)^4)/(
(2*c*d*x + b*d)^(9/2)*c^3*d^5)

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mupad [B]  time = 0.06, size = 67, normalized size = 0.76 \[ -\frac {{\left (b+2\,c\,x\right )}^4+\left (\frac {8\,a\,c}{5}-\frac {2\,b^2}{5}\right )\,{\left (b+2\,c\,x\right )}^2+\frac {b^4}{9}+\frac {16\,a^2\,c^2}{9}-\frac {8\,a\,b^2\,c}{9}}{16\,c^3\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(11/2),x)

[Out]

-((b + 2*c*x)^4 + ((8*a*c)/5 - (2*b^2)/5)*(b + 2*c*x)^2 + b^4/9 + (16*a^2*c^2)/9 - (8*a*b^2*c)/9)/(16*c^3*d*(b
*d + 2*c*d*x)^(9/2))

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sympy [A]  time = 14.65, size = 966, normalized size = 10.98 \[ \begin {cases} - \frac {5 a^{2} c^{2} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {2 a b^{2} c \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {18 a b c^{2} x \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {18 a c^{3} x^{2} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {2 b^{4} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {18 b^{3} c x \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {63 b^{2} c^{2} x^{2} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {90 b c^{3} x^{3} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} - \frac {45 c^{4} x^{4} \sqrt {b d + 2 c d x}}{45 b^{5} c^{3} d^{6} + 450 b^{4} c^{4} d^{6} x + 1800 b^{3} c^{5} d^{6} x^{2} + 3600 b^{2} c^{6} d^{6} x^{3} + 3600 b c^{7} d^{6} x^{4} + 1440 c^{8} d^{6} x^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{\left (b d\right )^{\frac {11}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**(11/2),x)

[Out]

Piecewise((-5*a**2*c**2*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x*
*2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 2*a*b**2*c*sqrt(b*d + 2*c*d*x)/
(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*
d**6*x**4 + 1440*c**8*d**6*x**5) - 18*a*b*c**2*x*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x
+ 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 18*a*c
**3*x**2*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*
c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 2*b**4*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6
+ 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c*
*8*d**6*x**5) - 18*b**3*c*x*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**
6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 63*b**2*c**2*x**2*sqrt(b*d
+ 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3
600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 90*b*c**3*x**3*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4
*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x*
*5) - 45*c**4*x**4*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 +
3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5), Ne(c, 0)), ((a**2*x + a*b*x**2 + b**2
*x**3/3)/(b*d)**(11/2), True))

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