3.19.11 \(\int \frac {(a+b x)^5}{(a c+(b c+a d) x+b d x^2)^2} \, dx\) [1811]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {640, 45} \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 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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 = 0.02, size = 114, normalized size = 1.52 \begin {gather*} -\frac {b^2 (2 b c-3 a d) x}{d^3}+\frac {b^3 x^2}{2 d^2}+\frac {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)}+\frac {3 \left (b^3 c^2-2 a b^2 c d+a^2 b d^2\right ) \log (c+d x)}{d^4} \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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Maple [A]
time = 0.73, size = 108, normalized size = 1.44

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{2} b d \,x^{2}+3 a d x -2 b c x \right )}{d^{3}}+\frac {3 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{d^{4} \left (d x +c \right )}\)	\(108\)
risch	\(\frac {b^{3} x^{2}}{2 d^{2}}+\frac {3 b^{2} a x}{d^{2}}-\frac {2 b^{3} c x}{d^{3}}+\frac {3 b \ln \left (d x +c \right ) a^{2}}{d^{2}}-\frac {6 b^{2} \ln \left (d x +c \right ) a c}{d^{3}}+\frac {3 b^{3} \ln \left (d x +c \right ) c^{2}}{d^{4}}-\frac {a^{3}}{d \left (d x +c \right )}+\frac {3 a^{2} b c}{d^{2} \left (d x +c \right )}-\frac {3 a \,b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {b^{3} c^{3}}{d^{4} \left (d x +c \right )}\)	\(149\)
norman	\(\frac {\frac {b^{4} x^{4}}{2 d}+\frac {b^{3} \left (7 a d -3 b c \right ) x^{3}}{2 d^{2}}-\frac {a \left (2 d^{3} a^{3} b +9 c^{2} d a \,b^{3}-6 b^{4} c^{3}\right )}{2 d^{4} b}-\frac {\left (8 a^{3} b^{2} d^{3}-3 a^{2} b^{3} c \,d^{2}+9 a \,b^{4} c^{2} d -6 b^{5} c^{3}\right ) x}{2 d^{4} b}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {3 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4}}\)	\(170\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (73) = 146\).
time = 2.15, 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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Sympy [A]
time = 0.29, 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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Giac [A]
time = 1.09, 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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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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