3.11.96 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^{13/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7} \]

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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} \begin {gather*} -\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(b^2 - 4*a*c)^3/(704*c^4*d*(b*d + 2*c*d*x)^(11/2)) - (3*(b^2 - 4*a*c)^2)/(448*c^4*d^3*(b*d + 2*c*d*x)^(7/2)) +
(b^2 - 4*a*c)/(64*c^4*d^5*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(64*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)^{13/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{13/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{9/2}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^4 (b d+2 c d x)^{5/2}}+\frac {1}{64 c^3 d^6 \sqrt {b d+2 c d x}}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 83, normalized size = 0.69 \begin {gather*} \frac {77 \left (b^2-4 a c\right ) (b+2 c x)^4-33 \left (b^2-4 a c\right )^2 (b+2 c x)^2+7 \left (b^2-4 a c\right )^3+77 (b+2 c x)^6}{4928 c^4 d (d (b+2 c x))^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(7*(b^2 - 4*a*c)^3 - 33*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 77*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 77*(b + 2*c*x)^6)/(49
28*c^4*d*(d*(b + 2*c*x))^(11/2))

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IntegrateAlgebraic [A]  time = 0.12, size = 174, normalized size = 1.44 \begin {gather*} \frac {-7 a^3 c^3-3 a^2 b^2 c^2-33 a^2 b c^3 x-33 a^2 c^4 x^2-2 a b^4 c-22 a b^3 c^2 x-99 a b^2 c^3 x^2-154 a b c^4 x^3-77 a c^5 x^4+2 b^6+22 b^5 c x+99 b^4 c^2 x^2+231 b^3 c^3 x^3+308 b^2 c^4 x^4+231 b c^5 x^5+77 c^6 x^6}{77 c^4 d (b d+2 c d x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(2*b^6 - 2*a*b^4*c - 3*a^2*b^2*c^2 - 7*a^3*c^3 + 22*b^5*c*x - 22*a*b^3*c^2*x - 33*a^2*b*c^3*x + 99*b^4*c^2*x^2
- 99*a*b^2*c^3*x^2 - 33*a^2*c^4*x^2 + 231*b^3*c^3*x^3 - 154*a*b*c^4*x^3 + 308*b^2*c^4*x^4 - 77*a*c^5*x^4 + 23
1*b*c^5*x^5 + 77*c^6*x^6)/(77*c^4*d*(b*d + 2*c*d*x)^(11/2))

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fricas [B]  time = 0.41, size = 252, normalized size = 2.08 \begin {gather*} \frac {{\left (77 \, c^{6} x^{6} + 231 \, b c^{5} x^{5} + 2 \, b^{6} - 2 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 77 \, {\left (4 \, b^{2} c^{4} - a c^{5}\right )} x^{4} + 77 \, {\left (3 \, b^{3} c^{3} - 2 \, a b c^{4}\right )} x^{3} + 33 \, {\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - a^{2} c^{4}\right )} x^{2} + 11 \, {\left (2 \, b^{5} c - 2 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{77 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} \end {gather*}

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)^(13/2),x, algorithm="fricas")

[Out]

1/77*(77*c^6*x^6 + 231*b*c^5*x^5 + 2*b^6 - 2*a*b^4*c - 3*a^2*b^2*c^2 - 7*a^3*c^3 + 77*(4*b^2*c^4 - a*c^5)*x^4
+ 77*(3*b^3*c^3 - 2*a*b*c^4)*x^3 + 33*(3*b^4*c^2 - 3*a*b^2*c^3 - a^2*c^4)*x^2 + 11*(2*b^5*c - 2*a*b^3*c^2 - 3*
a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(64*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7
*x^3 + 60*b^4*c^6*d^7*x^2 + 12*b^5*c^5*d^7*x + b^6*c^4*d^7)

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giac [A]  time = 0.28, size = 176, normalized size = 1.45 \begin {gather*} \frac {\sqrt {2 \, c d x + b d}}{64 \, c^{4} d^{7}} + \frac {7 \, b^{6} d^{4} - 84 \, a b^{4} c d^{4} + 336 \, a^{2} b^{2} c^{2} d^{4} - 448 \, a^{3} c^{3} d^{4} - 33 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 264 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 528 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 77 \, {\left (2 \, c d x + b d\right )}^{4} b^{2} - 308 \, {\left (2 \, c d x + b d\right )}^{4} a c}{4928 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{4} d^{5}} \end {gather*}

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)^(13/2),x, algorithm="giac")

[Out]

1/64*sqrt(2*c*d*x + b*d)/(c^4*d^7) + 1/4928*(7*b^6*d^4 - 84*a*b^4*c*d^4 + 336*a^2*b^2*c^2*d^4 - 448*a^3*c^3*d^
4 - 33*(2*c*d*x + b*d)^2*b^4*d^2 + 264*(2*c*d*x + b*d)^2*a*b^2*c*d^2 - 528*(2*c*d*x + b*d)^2*a^2*c^2*d^2 + 77*
(2*c*d*x + b*d)^4*b^2 - 308*(2*c*d*x + b*d)^4*a*c)/((2*c*d*x + b*d)^(11/2)*c^4*d^5)

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maple [A]  time = 0.05, size = 174, normalized size = 1.44 \begin {gather*} -\frac {\left (2 c x +b \right ) \left (-77 c^{6} x^{6}-231 b \,c^{5} x^{5}+77 a \,c^{5} x^{4}-308 b^{2} c^{4} x^{4}+154 a b \,c^{4} x^{3}-231 b^{3} c^{3} x^{3}+33 a^{2} c^{4} x^{2}+99 a \,b^{2} c^{3} x^{2}-99 b^{4} c^{2} x^{2}+33 a^{2} b \,c^{3} x +22 a \,b^{3} c^{2} x -22 b^{5} c x +7 a^{3} c^{3}+3 a^{2} b^{2} c^{2}+2 a \,b^{4} c -2 b^{6}\right )}{77 \left (2 c d x +b d \right )^{\frac {13}{2}} c^{4}} \end {gather*}

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)^(13/2),x)

[Out]

-1/77*(2*c*x+b)*(-77*c^6*x^6-231*b*c^5*x^5+77*a*c^5*x^4-308*b^2*c^4*x^4+154*a*b*c^4*x^3-231*b^3*c^3*x^3+33*a^2
*c^4*x^2+99*a*b^2*c^3*x^2-99*b^4*c^2*x^2+33*a^2*b*c^3*x+22*a*b^3*c^2*x-22*b^5*c*x+7*a^3*c^3+3*a^2*b^2*c^2+2*a*
b^4*c-2*b^6)/c^4/(2*c*d*x+b*d)^(13/2)

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maxima [A]  time = 1.43, size = 138, normalized size = 1.14 \begin {gather*} \frac {\frac {77 \, \sqrt {2 \, c d x + b d}}{c^{3} d^{6}} + \frac {77 \, {\left (2 \, c d x + b d\right )}^{4} {\left (b^{2} - 4 \, a c\right )} - 33 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} d^{2} + 7 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{4}}{{\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{3} d^{4}}}{4928 \, c d} \end {gather*}

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)^(13/2),x, algorithm="maxima")

[Out]

1/4928*(77*sqrt(2*c*d*x + b*d)/(c^3*d^6) + (77*(2*c*d*x + b*d)^4*(b^2 - 4*a*c) - 33*(b^4 - 8*a*b^2*c + 16*a^2*
c^2)*(2*c*d*x + b*d)^2*d^2 + 7*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^4)/((2*c*d*x + b*d)^(11/2)*c
^3*d^4))/(c*d)

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mupad [B]  time = 0.08, size = 147, normalized size = 1.21 \begin {gather*} \frac {\sqrt {b\,d+2\,c\,d\,x}}{64\,c^4\,d^7}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (\frac {48\,a^2\,c^2\,d^2}{7}-\frac {24\,a\,b^2\,c\,d^2}{7}+\frac {3\,b^4\,d^2}{7}\right )+{\left (b\,d+2\,c\,d\,x\right )}^4\,\left (4\,a\,c-b^2\right )-\frac {b^6\,d^4}{11}+\frac {64\,a^3\,c^3\,d^4}{11}-\frac {48\,a^2\,b^2\,c^2\,d^4}{11}+\frac {12\,a\,b^4\,c\,d^4}{11}}{64\,c^4\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{11/2}} \end {gather*}

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)^(13/2),x)

[Out]

(b*d + 2*c*d*x)^(1/2)/(64*c^4*d^7) - ((b*d + 2*c*d*x)^2*((3*b^4*d^2)/7 + (48*a^2*c^2*d^2)/7 - (24*a*b^2*c*d^2)
/7) + (b*d + 2*c*d*x)^4*(4*a*c - b^2) - (b^6*d^4)/11 + (64*a^3*c^3*d^4)/11 - (48*a^2*b^2*c^2*d^4)/11 + (12*a*b
^4*c*d^4)/11)/(64*c^4*d^5*(b*d + 2*c*d*x)^(11/2))

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sympy [A]  time = 33.38, size = 1975, normalized size = 16.32

result too large to display

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)**(13/2),x)

[Out]

Piecewise((-7*a**3*c**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x*
*2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) -
3*a**2*b**2*c**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12
320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2
*b*c**3*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**
3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2*c**4*x
**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7
*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 2*a*b**4*c*sqrt(b*d
+ 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 +
18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 22*a*b**3*c**2*x*sqrt(b*d + 2*c*d
*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b
**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 99*a*b**2*c**3*x**2*sqrt(b*d + 2*c*d*x)/
(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*
c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 154*a*b*c**4*x**3*sqrt(b*d + 2*c*d*x)/(77*b*
*6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d
**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 77*a*c**5*x**4*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d
**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4
+ 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 2*b**6*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c*
*5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d*
*7*x**5 + 4928*c**10*d**7*x**6) + 22*b**5*c*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x +
4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 49
28*c**10*d**7*x**6) + 99*b**4*c**2*x**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b
**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**
10*d**7*x**6) + 231*b**3*c**3*x**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c
**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d*
*7*x**6) + 308*b**2*c**4*x**4*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d
**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x*
*6) + 231*b*c**5*x**5*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2
+ 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 77
*c**6*x**6*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b*
*3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6), 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)**(13/2), True))

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