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

Optimal. Leaf size=130 \[ -\frac {5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac {5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac {10 b^2 x (b c-a d)^3}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac {b^5 (c+d x)^4}{4 d^6} \]

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Rubi [A]  time = 0.14, antiderivative size = 130, 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} \begin {gather*} -\frac {5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac {5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac {10 b^2 x (b c-a d)^3}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac {b^5 (c+d x)^4}{4 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(c + d*x)^2,x]

[Out]

(-10*b^2*(b*c - a*d)^3*x)/d^5 + (b*c - a*d)^5/(d^6*(c + d*x)) + (5*b^3*(b*c - a*d)^2*(c + d*x)^2)/d^6 - (5*b^4
*(b*c - a*d)*(c + d*x)^3)/(3*d^6) + (b^5*(c + d*x)^4)/(4*d^6) + (5*b*(b*c - a*d)^4*Log[c + d*x])/d^6

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}{(c+d x)^2} \, dx &=\int \left (-\frac {10 b^2 (b c-a d)^3}{d^5}+\frac {(-b c+a d)^5}{d^5 (c+d x)^2}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)}+\frac {10 b^3 (b c-a d)^2 (c+d x)}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^2}{d^5}+\frac {b^5 (c+d x)^3}{d^5}\right ) \, dx\\ &=-\frac {10 b^2 (b c-a d)^3 x}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b^3 (b c-a d)^2 (c+d x)^2}{d^6}-\frac {5 b^4 (b c-a d) (c+d x)^3}{3 d^6}+\frac {b^5 (c+d x)^4}{4 d^6}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 228, normalized size = 1.75 \begin {gather*} \frac {-12 a^5 d^5+60 a^4 b c d^4+120 a^3 b^2 d^3 \left (-c^2+c d x+d^2 x^2\right )+60 a^2 b^3 d^2 \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )+20 a b^4 d \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )+60 b (c+d x) (b c-a d)^4 \log (c+d x)+b^5 \left (12 c^5-48 c^4 d x-30 c^3 d^2 x^2+10 c^2 d^3 x^3-5 c d^4 x^4+3 d^5 x^5\right )}{12 d^6 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(c + d*x)^2,x]

[Out]

(60*a^4*b*c*d^4 - 12*a^5*d^5 + 120*a^3*b^2*d^3*(-c^2 + c*d*x + d^2*x^2) + 60*a^2*b^3*d^2*(2*c^3 - 4*c^2*d*x -
3*c*d^2*x^2 + d^3*x^3) + 20*a*b^4*d*(-3*c^4 + 9*c^3*d*x + 6*c^2*d^2*x^2 - 2*c*d^3*x^3 + d^4*x^4) + b^5*(12*c^5
- 48*c^4*d*x - 30*c^3*d^2*x^2 + 10*c^2*d^3*x^3 - 5*c*d^4*x^4 + 3*d^5*x^5) + 60*b*(b*c - a*d)^4*(c + d*x)*Log[
c + d*x])/(12*d^6*(c + d*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{(c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.21, size = 373, normalized size = 2.87 \begin {gather*} \frac {3 \, b^{5} d^{5} x^{5} + 12 \, b^{5} c^{5} - 60 \, a b^{4} c^{4} d + 120 \, a^{2} b^{3} c^{3} d^{2} - 120 \, a^{3} b^{2} c^{2} d^{3} + 60 \, a^{4} b c d^{4} - 12 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 4 \, a b^{4} d^{5}\right )} x^{4} + 10 \, {\left (b^{5} c^{2} d^{3} - 4 \, a b^{4} c d^{4} + 6 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} - 4 \, a^{3} b^{2} d^{5}\right )} x^{2} - 12 \, {\left (4 \, b^{5} c^{4} d - 15 \, a b^{4} c^{3} d^{2} + 20 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4}\right )} x + 60 \, {\left (b^{5} c^{5} - 4 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x\right )} \log \left (d x + c\right )}{12 \, {\left (d^{7} x + c d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^5*d^5*x^5 + 12*b^5*c^5 - 60*a*b^4*c^4*d + 120*a^2*b^3*c^3*d^2 - 120*a^3*b^2*c^2*d^3 + 60*a^4*b*c*d^4
- 12*a^5*d^5 - 5*(b^5*c*d^4 - 4*a*b^4*d^5)*x^4 + 10*(b^5*c^2*d^3 - 4*a*b^4*c*d^4 + 6*a^2*b^3*d^5)*x^3 - 30*(b
^5*c^3*d^2 - 4*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 - 4*a^3*b^2*d^5)*x^2 - 12*(4*b^5*c^4*d - 15*a*b^4*c^3*d^2 + 20*
a^2*b^3*c^2*d^3 - 10*a^3*b^2*c*d^4)*x + 60*(b^5*c^5 - 4*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 - 4*a^3*b^2*c^2*d^3 +
a^4*b*c*d^4 + (b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)*log(d*x + c))
/(d^7*x + c*d^6)

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giac [B]  time = 1.27, size = 339, normalized size = 2.61 \begin {gather*} \frac {{\left (3 \, b^{5} - \frac {20 \, {\left (b^{5} c d - a b^{4} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac {60 \, {\left (b^{5} c^{2} d^{2} - 2 \, a b^{4} c d^{3} + a^{2} b^{3} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}} - \frac {120 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{{\left (d x + c\right )}^{3} d^{3}}\right )} {\left (d x + c\right )}^{4}}{12 \, d^{6}} - \frac {5 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{6}} + \frac {\frac {b^{5} c^{5} d^{4}}{d x + c} - \frac {5 \, a b^{4} c^{4} d^{5}}{d x + c} + \frac {10 \, a^{2} b^{3} c^{3} d^{6}}{d x + c} - \frac {10 \, a^{3} b^{2} c^{2} d^{7}}{d x + c} + \frac {5 \, a^{4} b c d^{8}}{d x + c} - \frac {a^{5} d^{9}}{d x + c}}{d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^5 - 20*(b^5*c*d - a*b^4*d^2)/((d*x + c)*d) + 60*(b^5*c^2*d^2 - 2*a*b^4*c*d^3 + a^2*b^3*d^4)/((d*x +
c)^2*d^2) - 120*(b^5*c^3*d^3 - 3*a*b^4*c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)/((d*x + c)^3*d^3))*(d*x + c)^4
/d^6 - 5*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*log(abs(d*x + c)/((d*x +
c)^2*abs(d)))/d^6 + (b^5*c^5*d^4/(d*x + c) - 5*a*b^4*c^4*d^5/(d*x + c) + 10*a^2*b^3*c^3*d^6/(d*x + c) - 10*a^3
*b^2*c^2*d^7/(d*x + c) + 5*a^4*b*c*d^8/(d*x + c) - a^5*d^9/(d*x + c))/d^10

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maple [B]  time = 0.01, size = 326, normalized size = 2.51 \begin {gather*} \frac {b^{5} x^{4}}{4 d^{2}}+\frac {5 a \,b^{4} x^{3}}{3 d^{2}}-\frac {2 b^{5} c \,x^{3}}{3 d^{3}}+\frac {5 a^{2} b^{3} x^{2}}{d^{2}}-\frac {5 a \,b^{4} c \,x^{2}}{d^{3}}+\frac {3 b^{5} c^{2} x^{2}}{2 d^{4}}-\frac {a^{5}}{\left (d x +c \right ) d}+\frac {5 a^{4} b c}{\left (d x +c \right ) d^{2}}+\frac {5 a^{4} b \ln \left (d x +c \right )}{d^{2}}-\frac {10 a^{3} b^{2} c^{2}}{\left (d x +c \right ) d^{3}}-\frac {20 a^{3} b^{2} c \ln \left (d x +c \right )}{d^{3}}+\frac {10 a^{3} b^{2} x}{d^{2}}+\frac {10 a^{2} b^{3} c^{3}}{\left (d x +c \right ) d^{4}}+\frac {30 a^{2} b^{3} c^{2} \ln \left (d x +c \right )}{d^{4}}-\frac {20 a^{2} b^{3} c x}{d^{3}}-\frac {5 a \,b^{4} c^{4}}{\left (d x +c \right ) d^{5}}-\frac {20 a \,b^{4} c^{3} \ln \left (d x +c \right )}{d^{5}}+\frac {15 a \,b^{4} c^{2} x}{d^{4}}+\frac {b^{5} c^{5}}{\left (d x +c \right ) d^{6}}+\frac {5 b^{5} c^{4} \ln \left (d x +c \right )}{d^{6}}-\frac {4 b^{5} c^{3} x}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^5/d^2*x^4+5/3*b^4/d^2*x^3*a-2/3*b^5/d^3*x^3*c+5*b^3/d^2*x^2*a^2-5*b^4/d^3*x^2*a*c+3/2*b^5/d^4*x^2*c^2+10
*b^2/d^2*a^3*x-20*b^3/d^3*a^2*c*x+15*b^4/d^4*a*c^2*x-4*b^5/d^5*c^3*x-1/d/(d*x+c)*a^5+5/d^2/(d*x+c)*a^4*b*c-10/
d^3/(d*x+c)*a^3*b^2*c^2+10/d^4/(d*x+c)*a^2*b^3*c^3-5/d^5/(d*x+c)*a*b^4*c^4+1/d^6/(d*x+c)*b^5*c^5+5*b/d^2*ln(d*
x+c)*a^4-20*b^2/d^3*ln(d*x+c)*a^3*c+30*b^3/d^4*ln(d*x+c)*a^2*c^2-20*b^4/d^5*ln(d*x+c)*a*c^3+5*b^5/d^6*ln(d*x+c
)*c^4

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maxima [B]  time = 1.40, size = 264, normalized size = 2.03 \begin {gather*} \frac {b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}}{d^{7} x + c d^{6}} + \frac {3 \, b^{5} d^{3} x^{4} - 4 \, {\left (2 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{3} + 6 \, {\left (3 \, b^{5} c^{2} d - 10 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{2} - 12 \, {\left (4 \, b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 20 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x}{12 \, d^{5}} + \frac {5 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (d x + c\right )}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)/(d^7*x + c*d^6)
+ 1/12*(3*b^5*d^3*x^4 - 4*(2*b^5*c*d^2 - 5*a*b^4*d^3)*x^3 + 6*(3*b^5*c^2*d - 10*a*b^4*c*d^2 + 10*a^2*b^3*d^3)*
x^2 - 12*(4*b^5*c^3 - 15*a*b^4*c^2*d + 20*a^2*b^3*c*d^2 - 10*a^3*b^2*d^3)*x)/d^5 + 5*(b^5*c^4 - 4*a*b^4*c^3*d
+ 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*log(d*x + c)/d^6

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mupad [B]  time = 0.25, size = 327, normalized size = 2.52 \begin {gather*} x^3\,\left (\frac {5\,a\,b^4}{3\,d^2}-\frac {2\,b^5\,c}{3\,d^3}\right )+x\,\left (\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d}-\frac {10\,a^2\,b^3}{d^2}+\frac {b^5\,c^2}{d^4}\right )}{d}+\frac {10\,a^3\,b^2}{d^2}-\frac {c^2\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d^2}\right )-x^2\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d}-\frac {5\,a^2\,b^3}{d^2}+\frac {b^5\,c^2}{2\,d^4}\right )+\frac {\ln \left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{d^6}-\frac {a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5}{d\,\left (x\,d^6+c\,d^5\right )}+\frac {b^5\,x^4}{4\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(c + d*x)^2,x)

[Out]

x^3*((5*a*b^4)/(3*d^2) - (2*b^5*c)/(3*d^3)) + x*((2*c*((2*c*((5*a*b^4)/d^2 - (2*b^5*c)/d^3))/d - (10*a^2*b^3)/
d^2 + (b^5*c^2)/d^4))/d + (10*a^3*b^2)/d^2 - (c^2*((5*a*b^4)/d^2 - (2*b^5*c)/d^3))/d^2) - x^2*((c*((5*a*b^4)/d
^2 - (2*b^5*c)/d^3))/d - (5*a^2*b^3)/d^2 + (b^5*c^2)/(2*d^4)) + (log(c + d*x)*(5*b^5*c^4 + 5*a^4*b*d^4 - 20*a^
3*b^2*c*d^3 + 30*a^2*b^3*c^2*d^2 - 20*a*b^4*c^3*d))/d^6 - (a^5*d^5 - b^5*c^5 - 10*a^2*b^3*c^3*d^2 + 10*a^3*b^2
*c^2*d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4)/(d*(c*d^5 + d^6*x)) + (b^5*x^4)/(4*d^2)

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sympy [A]  time = 0.89, size = 231, normalized size = 1.78 \begin {gather*} \frac {b^{5} x^{4}}{4 d^{2}} + \frac {5 b \left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{6}} + x^{3} \left (\frac {5 a b^{4}}{3 d^{2}} - \frac {2 b^{5} c}{3 d^{3}}\right ) + x^{2} \left (\frac {5 a^{2} b^{3}}{d^{2}} - \frac {5 a b^{4} c}{d^{3}} + \frac {3 b^{5} c^{2}}{2 d^{4}}\right ) + x \left (\frac {10 a^{3} b^{2}}{d^{2}} - \frac {20 a^{2} b^{3} c}{d^{3}} + \frac {15 a b^{4} c^{2}}{d^{4}} - \frac {4 b^{5} c^{3}}{d^{5}}\right ) + \frac {- a^{5} d^{5} + 5 a^{4} b c d^{4} - 10 a^{3} b^{2} c^{2} d^{3} + 10 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d + b^{5} c^{5}}{c d^{6} + d^{7} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**5*x**4/(4*d**2) + 5*b*(a*d - b*c)**4*log(c + d*x)/d**6 + x**3*(5*a*b**4/(3*d**2) - 2*b**5*c/(3*d**3)) + x**
2*(5*a**2*b**3/d**2 - 5*a*b**4*c/d**3 + 3*b**5*c**2/(2*d**4)) + x*(10*a**3*b**2/d**2 - 20*a**2*b**3*c/d**3 + 1
5*a*b**4*c**2/d**4 - 4*b**5*c**3/d**5) + (-a**5*d**5 + 5*a**4*b*c*d**4 - 10*a**3*b**2*c**2*d**3 + 10*a**2*b**3
*c**3*d**2 - 5*a*b**4*c**4*d + b**5*c**5)/(c*d**6 + d**7*x)

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