3.1335 \(\int \frac {(a+b x)^4}{c+d x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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)^4/(c + d*x),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 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)^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*}

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Mathematica [A]  time = 0.04, size = 115, normalized size = 1.17 \[ \frac {b d x \left (48 a^3 d^3+36 a^2 b d^2 (d x-2 c)+8 a b^2 d \left (6 c^2-3 c d x+2 d^2 x^2\right )+b^3 \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )+12 (b c-a d)^4 \log (c+d x)}{12 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x),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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fricas [A]  time = 0.45, size = 179, normalized size = 1.83 \[ \frac {3 \, b^{4} d^{4} x^{4} - 4 \, {\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c),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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giac [A]  time = 1.23, size = 184, normalized size = 1.88 \[ \frac {3 \, b^{4} d^{3} x^{4} - 4 \, b^{4} c d^{2} x^{3} + 16 \, a b^{3} d^{3} x^{3} + 6 \, b^{4} c^{2} d x^{2} - 24 \, a b^{3} c d^{2} x^{2} + 36 \, a^{2} b^{2} d^{3} x^{2} - 12 \, b^{4} c^{3} x + 48 \, a b^{3} c^{2} d x - 72 \, a^{2} b^{2} c d^{2} x + 48 \, a^{3} b d^{3} x}{12 \, d^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c),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

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maple [B]  time = 0.00, size = 209, normalized size = 2.13 \[ \frac {b^{4} x^{4}}{4 d}+\frac {4 a \,b^{3} x^{3}}{3 d}-\frac {b^{4} c \,x^{3}}{3 d^{2}}+\frac {3 a^{2} b^{2} x^{2}}{d}-\frac {2 a \,b^{3} c \,x^{2}}{d^{2}}+\frac {b^{4} c^{2} x^{2}}{2 d^{3}}+\frac {a^{4} \ln \left (d x +c \right )}{d}-\frac {4 a^{3} b c \ln \left (d x +c \right )}{d^{2}}+\frac {4 a^{3} b x}{d}+\frac {6 a^{2} b^{2} c^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {6 a^{2} b^{2} c x}{d^{2}}-\frac {4 a \,b^{3} c^{3} \ln \left (d x +c \right )}{d^{4}}+\frac {4 a \,b^{3} c^{2} x}{d^{3}}+\frac {b^{4} c^{4} \ln \left (d x +c \right )}{d^{5}}-\frac {b^{4} c^{3} x}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(d*x+c),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*a^2*c*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 = 1.41, size = 177, normalized size = 1.81 \[ \frac {3 \, b^{4} d^{3} x^{4} - 4 \, {\left (b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + 6 \, a^{2} b^{2} d^{3}\right )} x^{2} - 12 \, {\left (b^{4} c^{3} - 4 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{12 \, d^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(d*x+c),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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mupad [B]  time = 0.22, size = 189, normalized size = 1.93 \[ x^3\,\left (\frac {4\,a\,b^3}{3\,d}-\frac {b^4\,c}{3\,d^2}\right )+x\,\left (\frac {4\,a^3\,b}{d}+\frac {c\,\left (\frac {c\,\left (\frac {4\,a\,b^3}{d}-\frac {b^4\,c}{d^2}\right )}{d}-\frac {6\,a^2\,b^2}{d}\right )}{d}\right )-x^2\,\left (\frac {c\,\left (\frac {4\,a\,b^3}{d}-\frac {b^4\,c}{d^2}\right )}{2\,d}-\frac {3\,a^2\,b^2}{d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d^5}+\frac {b^4\,x^4}{4\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(c + d*x),x)

[Out]

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

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sympy [A]  time = 0.39, size = 136, normalized size = 1.39 \[ \frac {b^{4} x^{4}}{4 d} + x^{3} \left (\frac {4 a b^{3}}{3 d} - \frac {b^{4} c}{3 d^{2}}\right ) + x^{2} \left (\frac {3 a^{2} b^{2}}{d} - \frac {2 a b^{3} c}{d^{2}} + \frac {b^{4} c^{2}}{2 d^{3}}\right ) + x \left (\frac {4 a^{3} b}{d} - \frac {6 a^{2} b^{2} c}{d^{2}} + \frac {4 a b^{3} c^{2}}{d^{3}} - \frac {b^{4} c^{3}}{d^{4}}\right ) + \frac {\left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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