3.11.93 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^{7/2}} \, dx\)

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

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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 {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^4 d^5}-\frac {3 \left (b^2-4 a c\right )^2}{64 c^4 d^3 \sqrt {b d+2 c d x}}+\frac {\left (b^2-4 a c\right )^3}{320 c^4 d (b d+2 c d x)^{5/2}}+\frac {(b d+2 c d x)^{7/2}}{448 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)^(7/2),x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(7/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.12, size = 174, normalized size = 1.44 \begin {gather*} \frac {-7 a^3 c^3-21 a^2 b^2 c^2-105 a^2 b c^3 x-105 a^2 c^4 x^2+14 a b^4 c+70 a b^3 c^2 x+105 a b^2 c^3 x^2+70 a b c^4 x^3+35 a c^5 x^4-2 b^6-10 b^5 c x-15 b^4 c^2 x^2-5 b^3 c^3 x^3+10 b^2 c^4 x^4+15 b c^5 x^5+5 c^6 x^6}{35 c^4 d (b d+2 c d x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(7/2),x]

[Out]

(-2*b^6 + 14*a*b^4*c - 21*a^2*b^2*c^2 - 7*a^3*c^3 - 10*b^5*c*x + 70*a*b^3*c^2*x - 105*a^2*b*c^3*x - 15*b^4*c^2
*x^2 + 105*a*b^2*c^3*x^2 - 105*a^2*c^4*x^2 - 5*b^3*c^3*x^3 + 70*a*b*c^4*x^3 + 10*b^2*c^4*x^4 + 35*a*c^5*x^4 +
15*b*c^5*x^5 + 5*c^6*x^6)/(35*c^4*d*(b*d + 2*c*d*x)^(5/2))

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fricas [A]  time = 0.40, size = 208, normalized size = 1.72 \begin {gather*} \frac {{\left (5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} - 2 \, b^{6} + 14 \, a b^{4} c - 21 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 5 \, {\left (2 \, b^{2} c^{4} + 7 \, a c^{5}\right )} x^{4} - 5 \, {\left (b^{3} c^{3} - 14 \, a b c^{4}\right )} x^{3} - 15 \, {\left (b^{4} c^{2} - 7 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} - 5 \, {\left (2 \, b^{5} c - 14 \, a b^{3} c^{2} + 21 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{35 \, {\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\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)^(7/2),x, algorithm="fricas")

[Out]

1/35*(5*c^6*x^6 + 15*b*c^5*x^5 - 2*b^6 + 14*a*b^4*c - 21*a^2*b^2*c^2 - 7*a^3*c^3 + 5*(2*b^2*c^4 + 7*a*c^5)*x^4
- 5*(b^3*c^3 - 14*a*b*c^4)*x^3 - 15*(b^4*c^2 - 7*a*b^2*c^3 + 7*a^2*c^4)*x^2 - 5*(2*b^5*c - 14*a*b^3*c^2 + 21*
a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(8*c^7*d^4*x^3 + 12*b*c^6*d^4*x^2 + 6*b^2*c^5*d^4*x + b^3*c^4*d^4)

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giac [A]  time = 0.27, size = 186, normalized size = 1.54 \begin {gather*} \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} - 15 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} + 120 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 240 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{320 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{4} d^{3}} - \frac {7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{24} d^{44} - 28 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{25} d^{44} - {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{24} d^{42}}{448 \, c^{28} d^{49}} \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)^(7/2),x, algorithm="giac")

[Out]

1/320*(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 - 15*(2*c*d*x + b*d)^2*b^4 + 120*(2*c*d*
x + b*d)^2*a*b^2*c - 240*(2*c*d*x + b*d)^2*a^2*c^2)/((2*c*d*x + b*d)^(5/2)*c^4*d^3) - 1/448*(7*(2*c*d*x + b*d)
^(3/2)*b^2*c^24*d^44 - 28*(2*c*d*x + b*d)^(3/2)*a*c^25*d^44 - (2*c*d*x + b*d)^(7/2)*c^24*d^42)/(c^28*d^49)

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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 (-5 c^{6} x^{6}-15 b \,c^{5} x^{5}-35 a \,c^{5} x^{4}-10 b^{2} c^{4} x^{4}-70 a b \,c^{4} x^{3}+5 b^{3} c^{3} x^{3}+105 a^{2} c^{4} x^{2}-105 a \,b^{2} c^{3} x^{2}+15 b^{4} c^{2} x^{2}+105 a^{2} b \,c^{3} x -70 a \,b^{3} c^{2} x +10 b^{5} c x +7 a^{3} c^{3}+21 a^{2} b^{2} c^{2}-14 a \,b^{4} c +2 b^{6}\right )}{35 \left (2 c d x +b d \right )^{\frac {7}{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)^(7/2),x)

[Out]

-1/35*(2*c*x+b)*(-5*c^6*x^6-15*b*c^5*x^5-35*a*c^5*x^4-10*b^2*c^4*x^4-70*a*b*c^4*x^3+5*b^3*c^3*x^3+105*a^2*c^4*
x^2-105*a*b^2*c^3*x^2+15*b^4*c^2*x^2+105*a^2*b*c^3*x-70*a*b^3*c^2*x+10*b^5*c*x+7*a^3*c^3+21*a^2*b^2*c^2-14*a*b
^4*c+2*b^6)/c^4/(2*c*d*x+b*d)^(7/2)

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maxima [A]  time = 1.40, size = 142, normalized size = 1.17 \begin {gather*} -\frac {\frac {7 \, {\left (15 \, {\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 {5}{2}} c^{3} d^{2}} + \frac {5 \, {\left (7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )}}{c^{3} d^{6}}}{2240 \, 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)^(7/2),x, algorithm="maxima")

[Out]

-1/2240*(7*(15*(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)^(5/2)*c^3*d^2) + 5*(7*(2*c*d*x + b*d)^(3/2)*(b^2 - 4*a*c)*d^2 - (2*c*d*x + b*d)^(7/
2))/(c^3*d^6))/(c*d)

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mupad [B]  time = 0.52, size = 170, normalized size = 1.40 \begin {gather*} -\frac {7\,a^3\,c^3+21\,a^2\,b^2\,c^2+105\,a^2\,b\,c^3\,x+105\,a^2\,c^4\,x^2-14\,a\,b^4\,c-70\,a\,b^3\,c^2\,x-105\,a\,b^2\,c^3\,x^2-70\,a\,b\,c^4\,x^3-35\,a\,c^5\,x^4+2\,b^6+10\,b^5\,c\,x+15\,b^4\,c^2\,x^2+5\,b^3\,c^3\,x^3-10\,b^2\,c^4\,x^4-15\,b\,c^5\,x^5-5\,c^6\,x^6}{35\,c^4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}} \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)^(7/2),x)

[Out]

-(2*b^6 + 7*a^3*c^3 - 5*c^6*x^6 - 35*a*c^5*x^4 - 15*b*c^5*x^5 + 21*a^2*b^2*c^2 + 105*a^2*c^4*x^2 + 15*b^4*c^2*
x^2 + 5*b^3*c^3*x^3 - 10*b^2*c^4*x^4 - 14*a*b^4*c + 10*b^5*c*x - 105*a*b^2*c^3*x^2 - 70*a*b^3*c^2*x + 105*a^2*
b*c^3*x - 70*a*b*c^4*x^3)/(35*c^4*d*(b*d + 2*c*d*x)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(7/2),x)

[Out]

Timed out

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