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

Optimal. Leaf size=75 \[ -\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {(b c-a d)^3}{d^4 (c+d x)}+\frac {3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac {b^3 x^2}{2 d^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \begin {gather*} -\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {(b c-a d)^3}{d^4 (c+d x)}+\frac {3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac {b^3 x^2}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 114, normalized size = 1.52 \begin {gather*} \frac {3 \left (a^2 b d^2-2 a b^2 c d+b^3 c^2\right ) \log (c+d x)}{d^4}+\frac {-a^3 d^3+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3}{d^4 (c+d x)}-\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {b^3 x^2}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 172, normalized size = 2.29 \begin {gather*} \frac {b^{3} d^{3} x^{3} + 2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} - 3 \, {\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} x + 6 \, {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{5} x + c d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 118, normalized size = 1.57 \begin {gather*} \frac {3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4}} + \frac {b^{3} d^{2} x^{2} - 4 \, b^{3} c d x + 6 \, a b^{2} d^{2} x}{2 \, d^{4}} + \frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{{\left (d x + c\right )} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*log(abs(d*x + c))/d^4 + 1/2*(b^3*d^2*x^2 - 4*b^3*c*d*x + 6*a*b^2*d^2*x)/
d^4 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/((d*x + c)*d^4)

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maple [B]  time = 0.05, size = 149, normalized size = 1.99 \begin {gather*} \frac {b^{3} x^{2}}{2 d^{2}}-\frac {a^{3}}{\left (d x +c \right ) d}+\frac {3 a^{2} b c}{\left (d x +c \right ) d^{2}}+\frac {3 a^{2} b \ln \left (d x +c \right )}{d^{2}}-\frac {3 a \,b^{2} c^{2}}{\left (d x +c \right ) d^{3}}-\frac {6 a \,b^{2} c \ln \left (d x +c \right )}{d^{3}}+\frac {3 a \,b^{2} x}{d^{2}}+\frac {b^{3} c^{3}}{\left (d x +c \right ) d^{4}}+\frac {3 b^{3} c^{2} \ln \left (d x +c \right )}{d^{4}}-\frac {2 b^{3} c x}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.12, size = 117, normalized size = 1.56 \begin {gather*} \frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{d^{5} x + c d^{4}} + \frac {b^{3} d x^{2} - 2 \, {\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x}{2 \, d^{3}} + \frac {3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (d x + c\right )}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.59, size = 123, normalized size = 1.64 \begin {gather*} x\,\left (\frac {3\,a\,b^2}{d^2}-\frac {2\,b^3\,c}{d^3}\right )+\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{d^4}-\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{d\,\left (x\,d^4+c\,d^3\right )}+\frac {b^3\,x^2}{2\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.56, size = 102, normalized size = 1.36 \begin {gather*} \frac {b^{3} x^{2}}{2 d^{2}} + \frac {3 b \left (a d - b c\right )^{2} \log {\left (c + d x \right )}}{d^{4}} + x \left (\frac {3 a b^{2}}{d^{2}} - \frac {2 b^{3} c}{d^{3}}\right ) + \frac {- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}}{c d^{4} + d^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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