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

Optimal. Leaf size=117 \[ \frac{a^2 x (b c-a d)^2}{b^5}-\frac{a^3 (b c-a d)^2 \log (a+b x)}{b^6}+\frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{x^3 (b c-a d)^2}{3 b^3}-\frac{a x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.114562, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{a^2 x (b c-a d)^2}{b^5}-\frac{a^3 (b c-a d)^2 \log (a+b x)}{b^6}+\frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{x^3 (b c-a d)^2}{3 b^3}-\frac{a x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^2)/(a + b*x),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{x^3 (c+d x)^2}{a+b x} \, dx &=\int \left (\frac{a^2 (-b c+a d)^2}{b^5}-\frac{a (-b c+a d)^2 x}{b^4}+\frac{(b c-a d)^2 x^2}{b^3}+\frac{d (2 b c-a d) x^3}{b^2}+\frac{d^2 x^4}{b}-\frac{a^3 (-b c+a d)^2}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{a^2 (b c-a d)^2 x}{b^5}-\frac{a (b c-a d)^2 x^2}{2 b^4}+\frac{(b c-a d)^2 x^3}{3 b^3}+\frac{d (2 b c-a d) x^4}{4 b^2}+\frac{d^2 x^5}{5 b}-\frac{a^3 (b c-a d)^2 \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.060601, size = 112, normalized size = 0.96 \[ \frac{60 a^2 b x (b c-a d)^2-60 a^3 (b c-a d)^2 \log (a+b x)+15 b^4 d x^4 (2 b c-a d)+20 b^3 x^3 (b c-a d)^2-30 a b^2 x^2 (b c-a d)^2+12 b^5 d^2 x^5}{60 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a^2*b*(b*c - a*d)^2*x - 30*a*b^2*(b*c - a*d)^2*x^2 + 20*b^3*(b*c - a*d)^2*x^3 + 15*b^4*d*(2*b*c - a*d)*x^4
+ 12*b^5*d^2*x^5 - 60*a^3*(b*c - a*d)^2*Log[a + b*x])/(60*b^6)

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Maple [A]  time = 0.003, size = 192, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{5}}{5\,b}}-{\frac{{x}^{4}a{d}^{2}}{4\,{b}^{2}}}+{\frac{{x}^{4}cd}{2\,b}}+{\frac{{x}^{3}{a}^{2}{d}^{2}}{3\,{b}^{3}}}-{\frac{2\,{x}^{3}acd}{3\,{b}^{2}}}+{\frac{{x}^{3}{c}^{2}}{3\,b}}-{\frac{{x}^{2}{a}^{3}{d}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}{x}^{2}cd}{{b}^{3}}}-{\frac{a{x}^{2}{c}^{2}}{2\,{b}^{2}}}+{\frac{{a}^{4}{d}^{2}x}{{b}^{5}}}-2\,{\frac{{a}^{3}cdx}{{b}^{4}}}+{\frac{{a}^{2}{c}^{2}x}{{b}^{3}}}-{\frac{{a}^{5}\ln \left ( bx+a \right ){d}^{2}}{{b}^{6}}}+2\,{\frac{{a}^{4}\ln \left ( bx+a \right ) cd}{{b}^{5}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{2}}{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*d^2*x^5/b-1/4/b^2*x^4*a*d^2+1/2/b*x^4*c*d+1/3/b^3*x^3*a^2*d^2-2/3/b^2*x^3*a*c*d+1/3/b*x^3*c^2-1/2/b^4*x^2*
a^3*d^2+1/b^3*x^2*a^2*c*d-1/2/b^2*x^2*a*c^2+1/b^5*a^4*d^2*x-2/b^4*a^3*c*d*x+1/b^3*a^2*c^2*x-a^5/b^6*ln(b*x+a)*
d^2+2*a^4/b^5*ln(b*x+a)*c*d-a^3/b^4*ln(b*x+a)*c^2

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Maxima [A]  time = 1.03842, size = 228, normalized size = 1.95 \begin{align*} \frac{12 \, b^{4} d^{2} x^{5} + 15 \,{\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{4} + 20 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{3} - 30 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 60 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} x}{60 \, b^{5}} - \frac{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*b^4*d^2*x^5 + 15*(2*b^4*c*d - a*b^3*d^2)*x^4 + 20*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^3 - 30*(a*b
^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2 + 60*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*x)/b^5 - (a^3*b^2*c^2 - 2*a
^4*b*c*d + a^5*d^2)*log(b*x + a)/b^6

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Fricas [A]  time = 1.88777, size = 355, normalized size = 3.03 \begin{align*} \frac{12 \, b^{5} d^{2} x^{5} + 15 \,{\left (2 \, b^{5} c d - a b^{4} d^{2}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} - 30 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 60 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x - 60 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*b^5*d^2*x^5 + 15*(2*b^5*c*d - a*b^4*d^2)*x^4 + 20*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*x^3 - 30*(a*b
^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2 + 60*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*x - 60*(a^3*b^2*c^2 -
2*a^4*b*c*d + a^5*d^2)*log(b*x + a))/b^6

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Sympy [A]  time = 0.70332, size = 148, normalized size = 1.26 \begin{align*} - \frac{a^{3} \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{6}} + \frac{d^{2} x^{5}}{5 b} - \frac{x^{4} \left (a d^{2} - 2 b c d\right )}{4 b^{2}} + \frac{x^{3} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 b^{3}} - \frac{x^{2} \left (a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}\right )}{2 b^{4}} + \frac{x \left (a^{4} d^{2} - 2 a^{3} b c d + a^{2} b^{2} c^{2}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18902, size = 244, normalized size = 2.09 \begin{align*} \frac{12 \, b^{4} d^{2} x^{5} + 30 \, b^{4} c d x^{4} - 15 \, a b^{3} d^{2} x^{4} + 20 \, b^{4} c^{2} x^{3} - 40 \, a b^{3} c d x^{3} + 20 \, a^{2} b^{2} d^{2} x^{3} - 30 \, a b^{3} c^{2} x^{2} + 60 \, a^{2} b^{2} c d x^{2} - 30 \, a^{3} b d^{2} x^{2} + 60 \, a^{2} b^{2} c^{2} x - 120 \, a^{3} b c d x + 60 \, a^{4} d^{2} x}{60 \, b^{5}} - \frac{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*b^4*d^2*x^5 + 30*b^4*c*d*x^4 - 15*a*b^3*d^2*x^4 + 20*b^4*c^2*x^3 - 40*a*b^3*c*d*x^3 + 20*a^2*b^2*d^2*
x^3 - 30*a*b^3*c^2*x^2 + 60*a^2*b^2*c*d*x^2 - 30*a^3*b*d^2*x^2 + 60*a^2*b^2*c^2*x - 120*a^3*b*c*d*x + 60*a^4*d
^2*x)/b^5 - (a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*log(abs(b*x + a))/b^6