### 3.39 $$\int \frac{(a+b x^2) (-a d+4 b c x+3 b d x^2)}{(c+d x)^2} \, dx$$

Optimal. Leaf size=17 $\frac{\left (a+b x^2\right )^2}{c+d x}$

[Out]

(a + b*x^2)^2/(c + d*x)

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Rubi [A]  time = 0.0319914, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.031, Rules used = {1590} $\frac{\left (a+b x^2\right )^2}{c+d x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a + b*x^2)^2/(c + d*x)

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

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

Mathematica [B]  time = 0.0263226, size = 62, normalized size = 3.65 $\frac{a^2 d^4+2 a b d^2 \left (c^2+c d x+d^2 x^2\right )+b^2 \left (c^3 d x+c^4+d^4 x^4\right )}{d^4 (c+d x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.049, size = 76, normalized size = 4.5 \begin{align*}{\frac{b \left ( b{d}^{2}{x}^{3}-bcd{x}^{2}+2\,a{d}^{2}x+b{c}^{2}x \right ) }{{d}^{3}}}-{\frac{-{a}^{2}{d}^{4}-2\,ab{c}^{2}{d}^{2}-{b}^{2}{c}^{4}}{{d}^{4} \left ( dx+c \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.990938, size = 111, normalized size = 6.53 \begin{align*} \frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{d^{5} x + c d^{4}} + \frac{b^{2} d^{2} x^{3} - b^{2} c d x^{2} +{\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} x}{d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(3*b*d*x^2+4*b*c*x-a*d)/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.5962, size = 153, normalized size = 9. \begin{align*} \frac{b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} + b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4} +{\left (b^{2} c^{3} d + 2 \, a b c d^{3}\right )} x}{d^{5} x + c d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(3*b*d*x^2+4*b*c*x-a*d)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.56482, size = 75, normalized size = 4.41 \begin{align*} - \frac{b^{2} c x^{2}}{d^{2}} + \frac{b^{2} x^{3}}{d} + \frac{a^{2} d^{4} + 2 a b c^{2} d^{2} + b^{2} c^{4}}{c d^{4} + d^{5} x} + \frac{x \left (2 a b d^{2} + b^{2} c^{2}\right )}{d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(3*b*d*x**2+4*b*c*x-a*d)/(d*x+c)**2,x)

[Out]

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

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Giac [B]  time = 1.17215, size = 150, normalized size = 8.82 \begin{align*} \frac{{\left (b^{2} - \frac{4 \, b^{2} c}{d x + c} + \frac{6 \, b^{2} c^{2}}{{\left (d x + c\right )}^{2}} + \frac{2 \, a b d^{2}}{{\left (d x + c\right )}^{2}}\right )}{\left (d x + c\right )}^{3}}{d^{4}} + \frac{\frac{b^{2} c^{4} d^{3}}{d x + c} + \frac{2 \, a b c^{2} d^{5}}{d x + c} + \frac{a^{2} d^{7}}{d x + c}}{d^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(3*b*d*x^2+4*b*c*x-a*d)/(d*x+c)^2,x, algorithm="giac")

[Out]

(b^2 - 4*b^2*c/(d*x + c) + 6*b^2*c^2/(d*x + c)^2 + 2*a*b*d^2/(d*x + c)^2)*(d*x + c)^3/d^4 + (b^2*c^4*d^3/(d*x
+ c) + 2*a*b*c^2*d^5/(d*x + c) + a^2*d^7/(d*x + c))/d^7