3.444 \(\int \frac{a x^2+b x^3}{c x^2+d x^3} \, dx\)

Optimal. Leaf size=26 \[ \frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0452603, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1593, 1584, 43} \[ \frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3)/(c*x^2 + d*x^3),x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A]  time = 0.0075781, size = 25, normalized size = 0.96 \[ \frac{(a d-b c) \log (c+d x)}{d^2}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3)/(c*x^2 + d*x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 1.2 \begin{align*}{\frac{bx}{d}}+{\frac{\ln \left ( dx+c \right ) a}{d}}-{\frac{\ln \left ( dx+c \right ) bc}{{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a*x^2)/(d*x^3+c*x^2),x)

[Out]

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

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Maxima [A]  time = 0.963804, size = 35, normalized size = 1.35 \begin{align*} \frac{b x}{d} - \frac{{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a*x^2)/(d*x^3+c*x^2),x, algorithm="maxima")

[Out]

b*x/d - (b*c - a*d)*log(d*x + c)/d^2

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Fricas [A]  time = 0.962319, size = 54, normalized size = 2.08 \begin{align*} \frac{b d x -{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a*x^2)/(d*x^3+c*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.301285, size = 20, normalized size = 0.77 \begin{align*} \frac{b x}{d} + \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a*x**2)/(d*x**3+c*x**2),x)

[Out]

b*x/d + (a*d - b*c)*log(c + d*x)/d**2

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Giac [A]  time = 1.13029, size = 36, normalized size = 1.38 \begin{align*} \frac{b x}{d} - \frac{{\left (b c - a d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a*x^2)/(d*x^3+c*x^2),x, algorithm="giac")

[Out]

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