### 3.1800 $$\int \frac{(a+b x)^5}{a c+(b c+a d) x+b d x^2} \, dx$$

Optimal. Leaf size=98 $-\frac{b x (b c-a d)^3}{d^4}+\frac{(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac{(a+b x)^3 (b c-a d)}{3 d^2}+\frac{(b c-a d)^4 \log (c+d x)}{d^5}+\frac{(a+b x)^4}{4 d}$

[Out]

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

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Rubi [A]  time = 0.046089, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 43} $-\frac{b x (b c-a d)^3}{d^4}+\frac{(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac{(a+b x)^3 (b c-a d)}{3 d^2}+\frac{(b c-a d)^4 \log (c+d x)}{d^5}+\frac{(a+b x)^4}{4 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0403413, size = 115, normalized size = 1.17 $\frac{b d x \left (36 a^2 b d^2 (d x-2 c)+48 a^3 d^3+8 a b^2 d \left (6 c^2-3 c d x+2 d^2 x^2\right )+b^3 \left (6 c^2 d x-12 c^3-4 c d^2 x^2+3 d^3 x^3\right )\right )+12 (b c-a d)^4 \log (c+d x)}{12 d^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*d*x*(48*a^3*d^3 + 36*a^2*b*d^2*(-2*c + d*x) + 8*a*b^2*d*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + b^3*(-12*c^3 + 6*c^
2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3)) + 12*(b*c - a*d)^4*Log[c + d*x])/(12*d^5)

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Maple [B]  time = 0.041, size = 209, normalized size = 2.1 \begin{align*}{\frac{{b}^{4}{x}^{4}}{4\,d}}+{\frac{4\,a{b}^{3}{x}^{3}}{3\,d}}-{\frac{{b}^{4}{x}^{3}c}{3\,{d}^{2}}}+3\,{\frac{{b}^{2}{x}^{2}{a}^{2}}{d}}-2\,{\frac{{b}^{3}{x}^{2}ac}{{d}^{2}}}+{\frac{{x}^{2}{b}^{4}{c}^{2}}{2\,{d}^{3}}}+4\,{\frac{x{a}^{3}b}{d}}-6\,{\frac{{b}^{2}c{a}^{2}x}{{d}^{2}}}+4\,{\frac{xa{b}^{3}{c}^{2}}{{d}^{3}}}-{\frac{{b}^{4}{c}^{3}x}{{d}^{4}}}+{\frac{\ln \left ( dx+c \right ){a}^{4}}{d}}-4\,{\frac{\ln \left ( dx+c \right ){a}^{3}bc}{{d}^{2}}}+6\,{\frac{\ln \left ( dx+c \right ){a}^{2}{b}^{2}{c}^{2}}{{d}^{3}}}-4\,{\frac{\ln \left ( dx+c \right ) a{b}^{3}{c}^{3}}{{d}^{4}}}+{\frac{\ln \left ( dx+c \right ){b}^{4}{c}^{4}}{{d}^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09025, size = 239, normalized size = 2.44 \begin{align*} \frac{3 \, b^{4} d^{3} x^{4} - 4 \,{\left (b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + 6 \, a^{2} b^{2} d^{3}\right )} x^{2} - 12 \,{\left (b^{4} c^{3} - 4 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{12 \, d^{4}} + \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*d^3*x^4 - 4*(b^4*c*d^2 - 4*a*b^3*d^3)*x^3 + 6*(b^4*c^2*d - 4*a*b^3*c*d^2 + 6*a^2*b^2*d^3)*x^2 - 12
*(b^4*c^3 - 4*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 - 4*a^3*b*d^3)*x)/d^4 + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d
^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(d*x + c)/d^5

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Fricas [A]  time = 1.6222, size = 369, normalized size = 3.77 \begin{align*} \frac{3 \, b^{4} d^{4} x^{4} - 4 \,{\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} - 12 \,{\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \, d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*d^4*x^4 - 4*(b^4*c*d^3 - 4*a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*x^2 -
12*(b^4*c^3*d - 4*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*x + 12*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c
^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(d*x + c))/d^5

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Sympy [A]  time = 0.891777, size = 134, normalized size = 1.37 \begin{align*} \frac{b^{4} x^{4}}{4 d} + \frac{x^{3} \left (4 a b^{3} d - b^{4} c\right )}{3 d^{2}} + \frac{x^{2} \left (6 a^{2} b^{2} d^{2} - 4 a b^{3} c d + b^{4} c^{2}\right )}{2 d^{3}} + \frac{x \left (4 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 4 a b^{3} c^{2} d - b^{4} c^{3}\right )}{d^{4}} + \frac{\left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26526, size = 248, normalized size = 2.53 \begin{align*} \frac{3 \, b^{4} d^{3} x^{4} - 4 \, b^{4} c d^{2} x^{3} + 16 \, a b^{3} d^{3} x^{3} + 6 \, b^{4} c^{2} d x^{2} - 24 \, a b^{3} c d^{2} x^{2} + 36 \, a^{2} b^{2} d^{3} x^{2} - 12 \, b^{4} c^{3} x + 48 \, a b^{3} c^{2} d x - 72 \, a^{2} b^{2} c d^{2} x + 48 \, a^{3} b d^{3} x}{12 \, d^{4}} + \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*d^3*x^4 - 4*b^4*c*d^2*x^3 + 16*a*b^3*d^3*x^3 + 6*b^4*c^2*d*x^2 - 24*a*b^3*c*d^2*x^2 + 36*a^2*b^2*d
^3*x^2 - 12*b^4*c^3*x + 48*a*b^3*c^2*d*x - 72*a^2*b^2*c*d^2*x + 48*a^3*b*d^3*x)/d^4 + (b^4*c^4 - 4*a*b^3*c^3*d
+ 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(abs(d*x + c))/d^5