3.1927 \(\int \frac{(a+b x) (d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=97 \[ \frac{e x (b d-a e)^3}{b^4}+\frac{(d+e x)^2 (b d-a e)^2}{2 b^3}+\frac{(d+e x)^3 (b d-a e)}{3 b^2}+\frac{(b d-a e)^4 \log (a+b x)}{b^5}+\frac{(d+e x)^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.041532, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{e x (b d-a e)^3}{b^4}+\frac{(d+e x)^2 (b d-a e)^2}{2 b^3}+\frac{(d+e x)^3 (b d-a e)}{3 b^2}+\frac{(b d-a e)^4 \log (a+b x)}{b^5}+\frac{(d+e x)^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 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) (d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^4}{a+b x} \, dx\\ &=\int \left (\frac{e (b d-a e)^3}{b^4}+\frac{(b d-a e)^4}{b^4 (a+b x)}+\frac{e (b d-a e)^2 (d+e x)}{b^3}+\frac{e (b d-a e) (d+e x)^2}{b^2}+\frac{e (d+e x)^3}{b}\right ) \, dx\\ &=\frac{e (b d-a e)^3 x}{b^4}+\frac{(b d-a e)^2 (d+e x)^2}{2 b^3}+\frac{(b d-a e) (d+e x)^3}{3 b^2}+\frac{(d+e x)^4}{4 b}+\frac{(b d-a e)^4 \log (a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0455249, size = 114, normalized size = 1.18 \[ \frac{b e x \left (6 a^2 b e^2 (8 d+e x)-12 a^3 e^3-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (a+b x)}{12 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(b*e*x*(-12*a^3*e^3 + 6*a^2*b*e^2*(8*d + e*x) - 4*a*b^2*e*(18*d^2 + 6*d*e*x + e^2*x^2) + b^3*(48*d^3 + 36*d^2*
e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + 12*(b*d - a*e)^4*Log[a + b*x])/(12*b^5)

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Maple [B]  time = 0.003, size = 209, normalized size = 2.2 \begin{align*}{\frac{{e}^{4}{x}^{4}}{4\,b}}-{\frac{{e}^{4}{x}^{3}a}{3\,{b}^{2}}}+{\frac{4\,{e}^{3}{x}^{3}d}{3\,b}}+{\frac{{e}^{4}{x}^{2}{a}^{2}}{2\,{b}^{3}}}-2\,{\frac{{e}^{3}{x}^{2}ad}{{b}^{2}}}+3\,{\frac{{e}^{2}{x}^{2}{d}^{2}}{b}}-{\frac{{e}^{4}{a}^{3}x}{{b}^{4}}}+4\,{\frac{{e}^{3}{a}^{2}dx}{{b}^{3}}}-6\,{\frac{a{d}^{2}{e}^{2}x}{{b}^{2}}}+4\,{\frac{e{d}^{3}x}{b}}+{\frac{\ln \left ( bx+a \right ){a}^{4}{e}^{4}}{{b}^{5}}}-4\,{\frac{\ln \left ( bx+a \right ){a}^{3}d{e}^{3}}{{b}^{4}}}+6\,{\frac{\ln \left ( bx+a \right ){a}^{2}{d}^{2}{e}^{2}}{{b}^{3}}}-4\,{\frac{\ln \left ( bx+a \right ) a{d}^{3}e}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{4}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.969775, size = 242, normalized size = 2.49 \begin{align*} \frac{3 \, b^{3} e^{4} x^{4} + 4 \,{\left (4 \, b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 6 \,{\left (6 \, b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + 12 \,{\left (4 \, b^{3} d^{3} e - 6 \, a b^{2} d^{2} e^{2} + 4 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} x}{12 \, b^{4}} + \frac{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47019, size = 369, normalized size = 3.8 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} + 4 \,{\left (4 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (6 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (4 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 4 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.574397, size = 134, normalized size = 1.38 \begin{align*} \frac{e^{4} x^{4}}{4 b} - \frac{x^{3} \left (a e^{4} - 4 b d e^{3}\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} e^{4} - 4 a b d e^{3} + 6 b^{2} d^{2} e^{2}\right )}{2 b^{3}} - \frac{x \left (a^{3} e^{4} - 4 a^{2} b d e^{3} + 6 a b^{2} d^{2} e^{2} - 4 b^{3} d^{3} e\right )}{b^{4}} + \frac{\left (a e - b d\right )^{4} \log{\left (a + b x \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Giac [A]  time = 1.1113, size = 235, normalized size = 2.42 \begin{align*} \frac{3 \, b^{3} x^{4} e^{4} + 16 \, b^{3} d x^{3} e^{3} + 36 \, b^{3} d^{2} x^{2} e^{2} + 48 \, b^{3} d^{3} x e - 4 \, a b^{2} x^{3} e^{4} - 24 \, a b^{2} d x^{2} e^{3} - 72 \, a b^{2} d^{2} x e^{2} + 6 \, a^{2} b x^{2} e^{4} + 48 \, a^{2} b d x e^{3} - 12 \, a^{3} x e^{4}}{12 \, b^{4}} + \frac{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

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