3.1157 \(\int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=61 \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4 \]

[Out]

(b^2 - 4*a*c)*d^5*(b + 2*c*x)^2 + (d^5*(b + 2*c*x)^4)/2 + (b^2 - 4*a*c)^2*d^5*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.0426205, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {692, 628} \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4 \]

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2),x]

[Out]

(b^2 - 4*a*c)*d^5*(b + 2*c*x)^2 + (d^5*(b + 2*c*x)^4)/2 + (b^2 - 4*a*c)^2*d^5*Log[a + b*x + c*x^2]

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx &=\frac{1}{2} d^5 (b+2 c x)^4+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right ) d^5 (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right ) d^5 (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4+\left (b^2-4 a c\right )^2 d^5 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0232375, size = 54, normalized size = 0.89 \[ d^5 \left (8 c x (b+c x) \left (c \left (c x^2-2 a\right )+b^2+b c x\right )+\left (b^2-4 a c\right )^2 \log (a+x (b+c x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2),x]

[Out]

d^5*(8*c*x*(b + c*x)*(b^2 + b*c*x + c*(-2*a + c*x^2)) + (b^2 - 4*a*c)^2*Log[a + x*(b + c*x)])

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Maple [B]  time = 0.041, size = 133, normalized size = 2.2 \begin{align*} 8\,{x}^{4}{c}^{4}{d}^{5}+16\,{x}^{3}b{c}^{3}{d}^{5}-16\,{x}^{2}a{c}^{3}{d}^{5}+16\,{x}^{2}{b}^{2}{c}^{2}{d}^{5}+16\,\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{2}{d}^{5}-8\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}c{d}^{5}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}{d}^{5}-16\,xab{c}^{2}{d}^{5}+8\,x{b}^{3}c{d}^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^5/(c*x^2+b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.58318, size = 134, normalized size = 2.2 \begin{align*} 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \,{\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.69205, size = 207, normalized size = 3.39 \begin{align*} 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \,{\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.874252, size = 99, normalized size = 1.62 \begin{align*} 16 b c^{3} d^{5} x^{3} + 8 c^{4} d^{5} x^{4} + d^{5} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 16 a c^{3} d^{5} + 16 b^{2} c^{2} d^{5}\right ) + x \left (- 16 a b c^{2} d^{5} + 8 b^{3} c d^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**5/(c*x**2+b*x+a),x)

[Out]

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

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Giac [A]  time = 1.20101, size = 159, normalized size = 2.61 \begin{align*}{\left (b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) + \frac{8 \,{\left (c^{8} d^{5} x^{4} + 2 \, b c^{7} d^{5} x^{3} + 2 \, b^{2} c^{6} d^{5} x^{2} - 2 \, a c^{7} d^{5} x^{2} + b^{3} c^{5} d^{5} x - 2 \, a b c^{6} d^{5} x\right )}}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

(b^4*d^5 - 8*a*b^2*c*d^5 + 16*a^2*c^2*d^5)*log(c*x^2 + b*x + a) + 8*(c^8*d^5*x^4 + 2*b*c^7*d^5*x^3 + 2*b^2*c^6
*d^5*x^2 - 2*a*c^7*d^5*x^2 + b^3*c^5*d^5*x - 2*a*b*c^6*d^5*x)/c^4