3.20.27 \(\int \frac {(a+b x) (d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx\) [1927]

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

[Out]

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

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Rubi [A]
time = 0.03, 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, 45} \begin {gather*} \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} \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 45

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 = 0.03, 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs. \(2(91)=182\).
time = 1.02, size = 188, normalized size = 1.94

method	result	size
default	\(-\frac {e \left (-\frac {b^{3} x^{4} e^{3}}{4}+\frac {\left (\left (a e -2 b d \right ) b^{2} e^{2}-2 b^{3} d \,e^{2}\right ) x^{3}}{3}+\frac {\left (2 \left (a e -2 b d \right ) b^{2} d e -b e \left (a^{2} e^{2}-2 a b d e +2 b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (a e -2 b d \right ) \left (a^{2} e^{2}-2 a b d e +2 b^{2} d^{2}\right ) x \right )}{b^{4}}+\frac {\left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(188\)
risch	\(\frac {e^{4} x^{4}}{4 b}-\frac {e^{4} a \,x^{3}}{3 b^{2}}+\frac {4 e^{3} d \,x^{3}}{3 b}-\frac {2 e^{3} a d \,x^{2}}{b^{2}}+\frac {3 e^{2} d^{2} x^{2}}{b}+\frac {e^{4} a^{2} x^{2}}{2 b^{3}}-\frac {e^{4} a^{3} x}{b^{4}}+\frac {4 e^{3} a^{2} d x}{b^{3}}-\frac {6 e^{2} a \,d^{2} x}{b^{2}}+\frac {4 e \,d^{3} x}{b}+\frac {\ln \left (b x +a \right ) e^{4} a^{4}}{b^{5}}-\frac {4 \ln \left (b x +a \right ) a^{3} d \,e^{3}}{b^{4}}+\frac {6 \ln \left (b x +a \right ) a^{2} d^{2} e^{2}}{b^{3}}-\frac {4 \ln \left (b x +a \right ) a \,d^{3} e}{b^{2}}+\frac {\ln \left (b x +a \right ) d^{4}}{b}\)	\(209\)
norman	\(\frac {\frac {e^{4} x^{5}}{4}-\frac {e \left (e^{3} a^{3}-4 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -8 b^{3} d^{3}\right ) x^{2}}{2 b^{3}}+\frac {e^{2} \left (a^{2} e^{2}-4 a b d e +18 b^{2} d^{2}\right ) x^{3}}{6 b^{2}}-\frac {e^{3} \left (a e -16 b d \right ) x^{4}}{12 b}-\frac {\left (e^{4} a^{5}-4 d \,e^{3} a^{4} b +6 d^{2} e^{2} a^{3} b^{2}-4 d^{3} e \,a^{2} b^{3}\right ) x}{a \,b^{4}}}{b x +a}+\frac {\left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(228\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 169, normalized size = 1.74 \begin {gather*} \frac {3 \, b^{3} x^{4} e^{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*x^4*e^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 = 3.51, size = 170, normalized size = 1.75 \begin {gather*} \frac {48 \, b^{4} d^{3} x e + {\left (3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x\right )} e^{4} + 8 \, {\left (2 \, b^{4} d x^{3} - 3 \, a b^{3} d x^{2} + 6 \, a^{2} b^{2} d x\right )} e^{3} + 36 \, {\left (b^{4} d^{2} x^{2} - 2 \, a b^{3} d^{2} x\right )} e^{2} + 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*(48*b^4*d^3*x*e + (3*b^4*x^4 - 4*a*b^3*x^3 + 6*a^2*b^2*x^2 - 12*a^3*b*x)*e^4 + 8*(2*b^4*d*x^3 - 3*a*b^3*d
*x^2 + 6*a^2*b^2*d*x)*e^3 + 36*(b^4*d^2*x^2 - 2*a*b^3*d^2*x)*e^2 + 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 = 0.22, 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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Giac [A]
time = 1.72, 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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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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