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

Optimal. Leaf size=37 \[ \frac {\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {682} \begin {gather*} \frac {\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )^2}{(b d+2 c d x)^7} \, dx &=\frac {\left (a+b x+c x^2\right )^3}{3 \left (b^2-4 a c\right ) d^7 (b+2 c x)^6}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 1.76 \begin {gather*} -\frac {16 a^2 c^2-3 \left (b^2-4 a c\right ) (b+2 c x)^2-8 a b^2 c+b^4+3 (b+2 c x)^4}{192 c^3 d^7 (b+2 c x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(b^4 - 8*a*b^2*c + 16*a^2*c^2 - 3*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 3*(b + 2*c*x)^4)/(c^3*d^7*(b + 2*c*x)^6
)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.39, size = 164, normalized size = 4.43 \begin {gather*} -\frac {48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \, {\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \end {gather*}

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)^7,x, algorithm="fricas")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + b^4 + 4*a*b^2*c + 16*a^2*c^2 + 12*(5*b^2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c +
4*a*b*c^2)*x)/(64*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7
*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)

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giac [B]  time = 0.18, size = 87, normalized size = 2.35 \begin {gather*} -\frac {48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 60 \, b^{2} c^{2} x^{2} + 48 \, a c^{3} x^{2} + 12 \, b^{3} c x + 48 \, a b c^{2} x + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2}}{192 \, {\left (2 \, c x + b\right )}^{6} c^{3} d^{7}} \end {gather*}

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)^7,x, algorithm="giac")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + 60*b^2*c^2*x^2 + 48*a*c^3*x^2 + 12*b^3*c*x + 48*a*b*c^2*x + b^4 + 4*a*b^2*
c + 16*a^2*c^2)/((2*c*x + b)^6*c^3*d^7)

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maple [B]  time = 0.05, size = 74, normalized size = 2.00 \begin {gather*} \frac {-\frac {4 a c -b^{2}}{64 \left (2 c x +b \right )^{4} c^{3}}-\frac {1}{64 \left (2 c x +b \right )^{2} c^{3}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{192 \left (2 c x +b \right )^{6} c^{3}}}{d^{7}} \end {gather*}

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)^7,x)

[Out]

1/d^7*(-1/64*(4*a*c-b^2)/c^3/(2*c*x+b)^4-1/64/c^3/(2*c*x+b)^2-1/192*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^6
)

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maxima [B]  time = 1.50, size = 164, normalized size = 4.43 \begin {gather*} -\frac {48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \, {\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \end {gather*}

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)^7,x, algorithm="maxima")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + b^4 + 4*a*b^2*c + 16*a^2*c^2 + 12*(5*b^2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c +
4*a*b*c^2)*x)/(64*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7
*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)

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mupad [B]  time = 0.48, size = 157, normalized size = 4.24 \begin {gather*} -\frac {\frac {16\,a^2\,c^2+4\,a\,b^2\,c+b^4}{192\,c^3}+\frac {b\,x^3}{2}+\frac {c\,x^4}{4}+\frac {x^2\,\left (5\,b^2+4\,a\,c\right )}{16\,c}+\frac {b\,x\,\left (b^2+4\,a\,c\right )}{16\,c^2}}{b^6\,d^7+12\,b^5\,c\,d^7\,x+60\,b^4\,c^2\,d^7\,x^2+160\,b^3\,c^3\,d^7\,x^3+240\,b^2\,c^4\,d^7\,x^4+192\,b\,c^5\,d^7\,x^5+64\,c^6\,d^7\,x^6} \end {gather*}

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)^7,x)

[Out]

-((b^4 + 16*a^2*c^2 + 4*a*b^2*c)/(192*c^3) + (b*x^3)/2 + (c*x^4)/4 + (x^2*(4*a*c + 5*b^2))/(16*c) + (b*x*(4*a*
c + b^2))/(16*c^2))/(b^6*d^7 + 64*c^6*d^7*x^6 + 192*b*c^5*d^7*x^5 + 60*b^4*c^2*d^7*x^2 + 160*b^3*c^3*d^7*x^3 +
240*b^2*c^4*d^7*x^4 + 12*b^5*c*d^7*x)

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sympy [B]  time = 2.37, size = 173, normalized size = 4.68 \begin {gather*} \frac {- 16 a^{2} c^{2} - 4 a b^{2} c - b^{4} - 96 b c^{3} x^{3} - 48 c^{4} x^{4} + x^{2} \left (- 48 a c^{3} - 60 b^{2} c^{2}\right ) + x \left (- 48 a b c^{2} - 12 b^{3} c\right )}{192 b^{6} c^{3} d^{7} + 2304 b^{5} c^{4} d^{7} x + 11520 b^{4} c^{5} d^{7} x^{2} + 30720 b^{3} c^{6} d^{7} x^{3} + 46080 b^{2} c^{7} d^{7} x^{4} + 36864 b c^{8} d^{7} x^{5} + 12288 c^{9} d^{7} x^{6}} \end {gather*}

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)**7,x)

[Out]

(-16*a**2*c**2 - 4*a*b**2*c - b**4 - 96*b*c**3*x**3 - 48*c**4*x**4 + x**2*(-48*a*c**3 - 60*b**2*c**2) + x*(-48
*a*b*c**2 - 12*b**3*c))/(192*b**6*c**3*d**7 + 2304*b**5*c**4*d**7*x + 11520*b**4*c**5*d**7*x**2 + 30720*b**3*c
**6*d**7*x**3 + 46080*b**2*c**7*d**7*x**4 + 36864*b*c**8*d**7*x**5 + 12288*c**9*d**7*x**6)

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