3.1182 $$\int \frac{(b d+2 c d x)^5}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=73 $16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}$

[Out]

-(d^5*(b + 2*c*x)^4)/(2*(a + b*x + c*x^2)^2) - (4*c*d^5*(b + 2*c*x)^2)/(a + b*x + c*x^2) + 16*c^2*d^5*Log[a +
b*x + c*x^2]

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Rubi [A]  time = 0.0379107, antiderivative size = 73, 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 = {686, 628} $16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^3,x]

[Out]

-(d^5*(b + 2*c*x)^4)/(2*(a + b*x + c*x^2)^2) - (4*c*d^5*(b + 2*c*x)^2)/(a + b*x + c*x^2) + 16*c^2*d^5*Log[a +
b*x + c*x^2]

Rule 686

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

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}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}+\left (4 c d^2\right ) \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}+\left (16 c^2 d^4\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}+16 c^2 d^5 \log \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0418596, size = 65, normalized size = 0.89 $d^5 \left (16 c^2 \log (a+x (b+c x))-\frac{\left (b^2-4 a c\right ) \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )}{2 (a+x (b+c x))^2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^3,x]

[Out]

d^5*(-((b^2 - 4*a*c)*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)))/(2*(a + x*(b + c*x))^2) + 16*c^2*Log[a + x*(b + c
*x)])

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Maple [B]  time = 0.048, size = 181, normalized size = 2.5 \begin{align*} 32\,{\frac{{x}^{2}a{c}^{3}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{{x}^{2}{b}^{2}{c}^{2}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+32\,{\frac{xab{c}^{2}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{x{b}^{3}c{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+24\,{\frac{{d}^{5}{a}^{2}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-4\,{\frac{{d}^{5}ac{b}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{5}{b}^{4}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+16\,{c}^{2}{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ) \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)^3,x)

[Out]

32*d^5/(c*x^2+b*x+a)^2*x^2*a*c^3-8*d^5/(c*x^2+b*x+a)^2*x^2*b^2*c^2+32*d^5/(c*x^2+b*x+a)^2*b*a*c^2*x-8*d^5/(c*x
^2+b*x+a)^2*b^3*c*x+24*d^5/(c*x^2+b*x+a)^2*a^2*c^2-4*d^5/(c*x^2+b*x+a)^2*a*c*b^2-1/2*d^5/(c*x^2+b*x+a)^2*b^4+1
6*c^2*d^5*ln(c*x^2+b*x+a)

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Maxima [A]  time = 1.2268, size = 167, normalized size = 2.29 \begin{align*} 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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)^3,x, algorithm="maxima")

[Out]

16*c^2*d^5*log(c*x^2 + b*x + a) - 1/2*(16*(b^2*c^2 - 4*a*c^3)*d^5*x^2 + 16*(b^3*c - 4*a*b*c^2)*d^5*x + (b^4 +
8*a*b^2*c - 48*a^2*c^2)*d^5)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Fricas [B]  time = 2.35211, size = 385, normalized size = 5.27 \begin{align*} -\frac{16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5} - 32 \,{\left (c^{4} d^{5} x^{4} + 2 \, b c^{3} d^{5} x^{3} + 2 \, a b c^{2} d^{5} x + a^{2} c^{2} d^{5} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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)^3,x, algorithm="fricas")

[Out]

-1/2*(16*(b^2*c^2 - 4*a*c^3)*d^5*x^2 + 16*(b^3*c - 4*a*b*c^2)*d^5*x + (b^4 + 8*a*b^2*c - 48*a^2*c^2)*d^5 - 32*
(c^4*d^5*x^4 + 2*b*c^3*d^5*x^3 + 2*a*b*c^2*d^5*x + a^2*c^2*d^5 + (b^2*c^2 + 2*a*c^3)*d^5*x^2)*log(c*x^2 + b*x
+ a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [B]  time = 6.81381, size = 141, normalized size = 1.93 \begin{align*} 16 c^{2} d^{5} \log{\left (a + b x + c x^{2} \right )} + \frac{48 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} - b^{4} d^{5} + x^{2} \left (64 a c^{3} d^{5} - 16 b^{2} c^{2} d^{5}\right ) + x \left (64 a b c^{2} d^{5} - 16 b^{3} c d^{5}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\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)**3,x)

[Out]

16*c**2*d**5*log(a + b*x + c*x**2) + (48*a**2*c**2*d**5 - 8*a*b**2*c*d**5 - b**4*d**5 + x**2*(64*a*c**3*d**5 -
16*b**2*c**2*d**5) + x*(64*a*b*c**2*d**5 - 16*b**3*c*d**5))/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x*
*2*(4*a*c + 2*b**2))

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Giac [A]  time = 1.1679, size = 149, normalized size = 2.04 \begin{align*} 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{b^{4} d^{5} + 8 \, a b^{2} c d^{5} - 48 \, a^{2} c^{2} d^{5} + 16 \,{\left (b^{2} c^{2} d^{5} - 4 \, a c^{3} d^{5}\right )} x^{2} + 16 \,{\left (b^{3} c d^{5} - 4 \, a b c^{2} d^{5}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

16*c^2*d^5*log(c*x^2 + b*x + a) - 1/2*(b^4*d^5 + 8*a*b^2*c*d^5 - 48*a^2*c^2*d^5 + 16*(b^2*c^2*d^5 - 4*a*c^3*d^
5)*x^2 + 16*(b^3*c*d^5 - 4*a*b*c^2*d^5)*x)/(c*x^2 + b*x + a)^2