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

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

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Rubi [A]  time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 114, normalized size = 1.18 \begin {gather*} \frac {b e x \left (-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (48 d^3+36 d^2 e x+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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 181, normalized size = 1.87 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.18, size = 174, normalized size = 1.79 \begin {gather*} \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 {gather*}

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

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maple [B]  time = 0.05, size = 209, normalized size = 2.15 \begin {gather*} \frac {e^{4} x^{4}}{4 b}-\frac {a \,e^{4} x^{3}}{3 b^{2}}+\frac {4 d \,e^{3} x^{3}}{3 b}+\frac {a^{2} e^{4} x^{2}}{2 b^{3}}-\frac {2 a d \,e^{3} x^{2}}{b^{2}}+\frac {3 d^{2} e^{2} x^{2}}{b}+\frac {a^{4} e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {4 a^{3} d \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {a^{3} e^{4} x}{b^{4}}+\frac {6 a^{2} d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {4 a^{2} d \,e^{3} x}{b^{3}}-\frac {4 a \,d^{3} e \ln \left (b x +a \right )}{b^{2}}-\frac {6 a \,d^{2} e^{2} x}{b^{2}}+\frac {d^{4} \ln \left (b x +a \right )}{b}+\frac {4 d^{3} e x}{b} \end {gather*}

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 = 0.48, size = 179, normalized size = 1.85 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 188, normalized size = 1.94 \begin {gather*} x\,\left (\frac {4\,d^3\,e}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,e^4}{b^2}-\frac {4\,d\,e^3}{b}\right )}{b}+\frac {6\,d^2\,e^2}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,e^4}{3\,b^2}-\frac {4\,d\,e^3}{3\,b}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,e^4}{b^2}-\frac {4\,d\,e^3}{b}\right )}{2\,b}+\frac {3\,d^2\,e^2}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{b^5}+\frac {e^4\,x^4}{4\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.44, size = 136, normalized size = 1.40 \begin {gather*} x^{3} \left (- \frac {a e^{4}}{3 b^{2}} + \frac {4 d e^{3}}{3 b}\right ) + x^{2} \left (\frac {a^{2} e^{4}}{2 b^{3}} - \frac {2 a d e^{3}}{b^{2}} + \frac {3 d^{2} e^{2}}{b}\right ) + x \left (- \frac {a^{3} e^{4}}{b^{4}} + \frac {4 a^{2} d e^{3}}{b^{3}} - \frac {6 a d^{2} e^{2}}{b^{2}} + \frac {4 d^{3} e}{b}\right ) + \frac {e^{4} x^{4}}{4 b} + \frac {\left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}

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]

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

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