### 3.41 $$\int \frac{(a+b x^2)^2 (-a d+6 b c x+5 b d x^2)}{(c+d x)^2} \, dx$$

Optimal. Leaf size=17 $\frac{\left (a+b x^2\right )^3}{c+d x}$

[Out]

(a + b*x^2)^3/(c + d*x)

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Rubi [A]  time = 0.0459761, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.029, Rules used = {1590} $\frac{\left (a+b x^2\right )^3}{c+d x}$

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x^2)^2*(-(a*d) + 6*b*c*x + 5*b*d*x^2))/(c + d*x)^2,x]

[Out]

(a + b*x^2)^3/(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 )^2 \left (-a d+6 b c x+5 b d x^2\right )}{(c+d x)^2} \, dx &=\frac{\left (a+b x^2\right )^3}{c+d x}\\ \end{align*}

Mathematica [B]  time = 0.0385234, size = 90, normalized size = 5.29 $\frac{3 a^2 b d^4 \left (c^2+c d x+d^2 x^2\right )+a^3 d^6+3 a b^2 d^2 \left (c^3 d x+c^4+d^4 x^4\right )+b^3 \left (c^5 d x+c^6+d^6 x^6\right )}{d^6 (c+d x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*x^2)^2*(-(a*d) + 6*b*c*x + 5*b*d*x^2))/(c + d*x)^2,x]

[Out]

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

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Maple [B]  time = 0.05, size = 157, normalized size = 9.2 \begin{align*}{\frac{b \left ({b}^{2}{d}^{4}{x}^{5}-{b}^{2}c{d}^{3}{x}^{4}+3\,ab{d}^{4}{x}^{3}+{b}^{2}{c}^{2}{d}^{2}{x}^{3}-3\,abc{d}^{3}{x}^{2}-{b}^{2}{c}^{3}d{x}^{2}+3\,{a}^{2}{d}^{4}x+3\,ab{c}^{2}{d}^{2}x+{b}^{2}{c}^{4}x \right ) }{{d}^{5}}}-{\frac{-{a}^{3}{d}^{6}-3\,{a}^{2}b{c}^{2}{d}^{4}-3\,a{b}^{2}{c}^{4}{d}^{2}-{b}^{3}{c}^{6}}{{d}^{6} \left ( dx+c \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(5*b*d*x^2+6*b*c*x-a*d)/(d*x+c)^2,x)

[Out]

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

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Maxima [B]  time = 1.00438, size = 216, normalized size = 12.71 \begin{align*} \frac{b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}}{d^{7} x + c d^{6}} + \frac{b^{3} d^{4} x^{5} - b^{3} c d^{3} x^{4} +{\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} d^{4}\right )} x^{3} -{\left (b^{3} c^{3} d + 3 \, a b^{2} c d^{3}\right )} x^{2} +{\left (b^{3} c^{4} + 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b d^{4}\right )} x}{d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(5*b*d*x^2+6*b*c*x-a*d)/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.68655, size = 234, normalized size = 13.76 \begin{align*} \frac{b^{3} d^{6} x^{6} + 3 \, a b^{2} d^{6} x^{4} + 3 \, a^{2} b d^{6} x^{2} + b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6} +{\left (b^{3} c^{5} d + 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c d^{5}\right )} x}{d^{7} x + c d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(5*b*d*x^2+6*b*c*x-a*d)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.777898, size = 155, normalized size = 9.12 \begin{align*} - \frac{b^{3} c x^{4}}{d^{2}} + \frac{b^{3} x^{5}}{d} + \frac{a^{3} d^{6} + 3 a^{2} b c^{2} d^{4} + 3 a b^{2} c^{4} d^{2} + b^{3} c^{6}}{c d^{6} + d^{7} x} + \frac{x^{3} \left (3 a b^{2} d^{2} + b^{3} c^{2}\right )}{d^{3}} - \frac{x^{2} \left (3 a b^{2} c d^{2} + b^{3} c^{3}\right )}{d^{4}} + \frac{x \left (3 a^{2} b d^{4} + 3 a b^{2} c^{2} d^{2} + b^{3} c^{4}\right )}{d^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(5*b*d*x**2+6*b*c*x-a*d)/(d*x+c)**2,x)

[Out]

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

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Giac [B]  time = 1.16157, size = 292, normalized size = 17.18 \begin{align*} \frac{{\left (b^{3} - \frac{6 \, b^{3} c}{d x + c} + \frac{15 \, b^{3} c^{2}}{{\left (d x + c\right )}^{2}} - \frac{20 \, b^{3} c^{3}}{{\left (d x + c\right )}^{3}} + \frac{15 \, b^{3} c^{4}}{{\left (d x + c\right )}^{4}} + \frac{3 \, a b^{2} d^{2}}{{\left (d x + c\right )}^{2}} - \frac{12 \, a b^{2} c d^{2}}{{\left (d x + c\right )}^{3}} + \frac{18 \, a b^{2} c^{2} d^{2}}{{\left (d x + c\right )}^{4}} + \frac{3 \, a^{2} b d^{4}}{{\left (d x + c\right )}^{4}}\right )}{\left (d x + c\right )}^{5}}{d^{6}} + \frac{\frac{b^{3} c^{6} d^{5}}{d x + c} + \frac{3 \, a b^{2} c^{4} d^{7}}{d x + c} + \frac{3 \, a^{2} b c^{2} d^{9}}{d x + c} + \frac{a^{3} d^{11}}{d x + c}}{d^{11}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(5*b*d*x^2+6*b*c*x-a*d)/(d*x+c)^2,x, algorithm="giac")

[Out]

(b^3 - 6*b^3*c/(d*x + c) + 15*b^3*c^2/(d*x + c)^2 - 20*b^3*c^3/(d*x + c)^3 + 15*b^3*c^4/(d*x + c)^4 + 3*a*b^2*
d^2/(d*x + c)^2 - 12*a*b^2*c*d^2/(d*x + c)^3 + 18*a*b^2*c^2*d^2/(d*x + c)^4 + 3*a^2*b*d^4/(d*x + c)^4)*(d*x +
c)^5/d^6 + (b^3*c^6*d^5/(d*x + c) + 3*a*b^2*c^4*d^7/(d*x + c) + 3*a^2*b*c^2*d^9/(d*x + c) + a^3*d^11/(d*x + c)
)/d^11