3.3.62 \(\int \frac {x (c+d x)^2}{(a+b x)^2} \, dx\) [262]

Optimal. Leaf size=77 \[ \frac {2 d (b c-a d) x}{b^3}+\frac {d^2 x^2}{2 b^2}+\frac {a (b c-a d)^2}{b^4 (a+b x)}+\frac {(b c-3 a d) (b c-a d) \log (a+b x)}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \begin {gather*} \frac {a (b c-a d)^2}{b^4 (a+b x)}+\frac {(b c-3 a d) (b c-a d) \log (a+b x)}{b^4}+\frac {2 d x (b c-a d)}{b^3}+\frac {d^2 x^2}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 81, normalized size = 1.05 \begin {gather*} \frac {4 b d (b c-a d) x+b^2 d^2 x^2+\frac {2 a (b c-a d)^2}{a+b x}+2 \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) \log (a+b x)}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 91, normalized size = 1.18

method	result	size
default	\(-\frac {d \left (-\frac {1}{2} b d \,x^{2}+2 a d x -2 b c x \right )}{b^{3}}+\frac {\left (3 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4}}+\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{4} \left (b x +a \right )}\)	\(91\)
norman	\(\frac {\frac {d^{2} x^{3}}{2 b}-\frac {d \left (3 a d -4 b c \right ) x^{2}}{2 b^{2}}-\frac {\left (3 a^{3} d^{2}-4 a^{2} b c d +a \,b^{2} c^{2}\right ) x}{b^{3} a}}{b x +a}+\frac {\left (3 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(106\)
risch	\(\frac {d^{2} x^{2}}{2 b^{2}}-\frac {2 d^{2} a x}{b^{3}}+\frac {2 d c x}{b^{2}}+\frac {3 \ln \left (b x +a \right ) a^{2} d^{2}}{b^{4}}-\frac {4 \ln \left (b x +a \right ) a c d}{b^{3}}+\frac {\ln \left (b x +a \right ) c^{2}}{b^{2}}+\frac {a^{3} d^{2}}{b^{4} \left (b x +a \right )}-\frac {2 a^{2} c d}{b^{3} \left (b x +a \right )}+\frac {a \,c^{2}}{b^{2} \left (b x +a \right )}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^2/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 99, normalized size = 1.29 \begin {gather*} \frac {a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}}{b^{5} x + a b^{4}} + \frac {b d^{2} x^{2} + 4 \, {\left (b c d - a d^{2}\right )} x}{2 \, b^{3}} + \frac {{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (75) = 150\).
time = 1.19, size = 152, normalized size = 1.97 \begin {gather*} \frac {b^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (4 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{2} + 4 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x + 2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} b c d + 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.26, size = 92, normalized size = 1.19 \begin {gather*} x \left (- \frac {2 a d^{2}}{b^{3}} + \frac {2 c d}{b^{2}}\right ) + \frac {a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}}{a b^{4} + b^{5} x} + \frac {d^{2} x^{2}}{2 b^{2}} + \frac {\left (a d - b c\right ) \left (3 a d - b c\right ) \log {\left (a + b x \right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.65, size = 149, normalized size = 1.94 \begin {gather*} \frac {\frac {{\left (d^{2} + \frac {2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x + a\right )} b}\right )} {\left (b x + a\right )}^{2}}{b^{3}} - \frac {2 \, {\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {2 \, {\left (\frac {a b^{4} c^{2}}{b x + a} - \frac {2 \, a^{2} b^{3} c d}{b x + a} + \frac {a^{3} b^{2} d^{2}}{b x + a}\right )}}{b^{5}}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 105, normalized size = 1.36 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{b^4}-x\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )+\frac {d^2\,x^2}{2\,b^2}+\frac {a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}{b\,\left (x\,b^4+a\,b^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + d*x)^2)/(a + b*x)^2,x)

[Out]

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

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