3.12.69 \(\int \frac {(b d+2 c d x)^5}{(a+b x+c x^2)^2} \, dx\) [1169]

Optimal. Leaf size=65 \[ 8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {700, 706, 642} \begin {gather*} 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 642

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]

Rule 700

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 706

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])

Rubi steps

\begin {align*} \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c d^2\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 64, normalized size = 0.98 \begin {gather*} d^5 \left (16 b c^2 x+16 c^3 x^2-\frac {\left (b^2-4 a c\right )^2}{a+x (b+c x)}+8 c \left (b^2-4 a c\right ) \log (a+x (b+c x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

d^5*(16*b*c^2*x + 16*c^3*x^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x)) + 8*c*(b^2 - 4*a*c)*Log[a + x*(b + c*x)])

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Maple [A]
time = 0.69, size = 78, normalized size = 1.20

method	result	size
default	\(d^{5} \left (16 c^{3} x^{2}+16 b \,c^{2} x -\frac {16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}{c \,x^{2}+b x +a}-8 c \left (4 a c -b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )\right )\)	\(78\)
risch	\(16 c^{3} d^{5} x^{2}+16 b \,c^{2} d^{5} x -\frac {16 d^{5} a^{2} c^{2}}{c \,x^{2}+b x +a}+\frac {8 d^{5} a c \,b^{2}}{c \,x^{2}+b x +a}-\frac {d^{5} b^{4}}{c \,x^{2}+b x +a}-32 \ln \left (c \,x^{2}+b x +a \right ) a \,c^{2} d^{5}+8 \ln \left (c \,x^{2}+b x +a \right ) b^{2} c \,d^{5}\)	\(128\)
norman	\(\frac {\frac {b \left (32 a^{2} c^{2} d^{5}-8 a \,b^{2} c \,d^{5}+b^{4} d^{5}\right ) x}{a}+\frac {c \left (32 a^{2} c^{2} d^{5}+8 a \,b^{2} c \,d^{5}+b^{4} d^{5}\right ) x^{2}}{a}+16 c^{4} d^{5} x^{4}+32 b \,c^{3} d^{5} x^{3}}{c \,x^{2}+b x +a}+\left (-32 d^{5} c^{2} a +8 b^{2} d^{5} c \right ) \ln \left (c \,x^{2}+b x +a \right )\)	\(142\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 86, normalized size = 1.32 \begin {gather*} 16 \, c^{3} d^{5} x^{2} + 16 \, b c^{2} d^{5} x + 8 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5}}{c x^{2} + b x + a} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (65) = 130\).
time = 1.60, size = 165, normalized size = 2.54 \begin {gather*} \frac {16 \, c^{4} d^{5} x^{4} + 32 \, b c^{3} d^{5} x^{3} + 16 \, a b c^{2} d^{5} x + 16 \, {\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} - {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} + 8 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{5}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.91, size = 90, normalized size = 1.38 \begin {gather*} 16 b c^{2} d^{5} x + 16 c^{3} d^{5} x^{2} - 8 c d^{5} \cdot \left (4 a c - b^{2}\right ) \log {\left (a + b x + c x^{2} \right )} + \frac {- 16 a^{2} c^{2} d^{5} + 8 a b^{2} c d^{5} - b^{4} d^{5}}{a + b x + c x^{2}} \end {gather*}

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)**2,x)

[Out]

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

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Giac [A]
time = 1.31, size = 100, normalized size = 1.54 \begin {gather*} 8 \, {\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}}{c x^{2} + b x + a} + \frac {16 \, {\left (c^{7} d^{5} x^{2} + b c^{6} d^{5} x\right )}}{c^{4}} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.49, size = 97, normalized size = 1.49 \begin {gather*} 16\,c^3\,d^5\,x^2-\frac {16\,a^2\,c^2\,d^5-8\,a\,b^2\,c\,d^5+b^4\,d^5}{c\,x^2+b\,x+a}-\ln \left (c\,x^2+b\,x+a\right )\,\left (32\,a\,c^2\,d^5-8\,b^2\,c\,d^5\right )+16\,b\,c^2\,d^5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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