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

Optimal. Leaf size=104 \[ -\frac {a^2 (b c-a d)^2}{b^5 (a+b x)}-\frac {2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac {x (b c-3 a d) (b c-a d)}{b^4}+\frac {d x^2 (b c-a d)}{b^3}+\frac {d^2 x^3}{3 b^2} \]

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Rubi [A]  time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a^2 (b c-a d)^2}{b^5 (a+b x)}+\frac {d x^2 (b c-a d)}{b^3}+\frac {x (b c-3 a d) (b c-a d)}{b^4}-\frac {2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac {d^2 x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.10, size = 114, normalized size = 1.10 \begin {gather*} \frac {3 b x \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-6 a \left (2 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (a+b x)-\frac {3 a^2 (b c-a d)^2}{a+b x}+3 b^2 d x^2 (b c-a d)+b^3 d^2 x^3}{3 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 (c+d x)^2}{(a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.01, size = 202, normalized size = 1.94 \begin {gather*} \frac {b^{4} d^{2} x^{4} - 3 \, a^{2} b^{2} c^{2} + 6 \, a^{3} b c d - 3 \, a^{4} d^{2} + {\left (3 \, b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{3} + 3 \, {\left (b^{4} c^{2} - 3 \, a b^{3} c d + 2 \, a^{2} b^{2} d^{2}\right )} x^{2} + 3 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x - 6 \, {\left (a^{2} b^{2} c^{2} - 3 \, a^{3} b c d + 2 \, a^{4} d^{2} + {\left (a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{3 \, {\left (b^{6} x + a b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.21, size = 188, normalized size = 1.81 \begin {gather*} \frac {{\left (d^{2} + \frac {3 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {3 \, {\left (b^{4} c^{2} - 6 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{3}}{3 \, b^{5}} + \frac {2 \, {\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5}} - \frac {\frac {a^{2} b^{5} c^{2}}{b x + a} - \frac {2 \, a^{3} b^{4} c d}{b x + a} + \frac {a^{4} b^{3} d^{2}}{b x + a}}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d^2 + 3*(b^2*c*d - 2*a*b*d^2)/((b*x + a)*b) + 3*(b^4*c^2 - 6*a*b^3*c*d + 6*a^2*b^2*d^2)/((b*x + a)^2*b^2)
)*(b*x + a)^3/b^5 + 2*(a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^5 - (a^2*
b^5*c^2/(b*x + a) - 2*a^3*b^4*c*d/(b*x + a) + a^4*b^3*d^2/(b*x + a))/b^8

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maple [A]  time = 0.01, size = 164, normalized size = 1.58 \begin {gather*} \frac {d^{2} x^{3}}{3 b^{2}}-\frac {a \,d^{2} x^{2}}{b^{3}}+\frac {c d \,x^{2}}{b^{2}}-\frac {a^{4} d^{2}}{\left (b x +a \right ) b^{5}}+\frac {2 a^{3} c d}{\left (b x +a \right ) b^{4}}-\frac {4 a^{3} d^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {a^{2} c^{2}}{\left (b x +a \right ) b^{3}}+\frac {6 a^{2} c d \ln \left (b x +a \right )}{b^{4}}+\frac {3 a^{2} d^{2} x}{b^{4}}-\frac {2 a \,c^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {4 a c d x}{b^{3}}+\frac {c^{2} x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.11, size = 138, normalized size = 1.33 \begin {gather*} -\frac {a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}}{b^{6} x + a b^{5}} + \frac {b^{2} d^{2} x^{3} + 3 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2} + 3 \, {\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x}{3 \, b^{4}} - \frac {2 \, {\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.35, size = 158, normalized size = 1.52 \begin {gather*} x\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )-x^2\,\left (\frac {a\,d^2}{b^3}-\frac {c\,d}{b^2}\right )-\frac {a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2}{b\,\left (x\,b^5+a\,b^4\right )}-\frac {\ln \left (a+b\,x\right )\,\left (4\,a^3\,d^2-6\,a^2\,b\,c\,d+2\,a\,b^2\,c^2\right )}{b^5}+\frac {d^2\,x^3}{3\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x)^2)/(a + b*x)^2,x)

[Out]

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

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sympy [A]  time = 0.86, size = 126, normalized size = 1.21 \begin {gather*} - \frac {2 a \left (a d - b c\right ) \left (2 a d - b c\right ) \log {\left (a + b x \right )}}{b^{5}} + x^{2} \left (- \frac {a d^{2}}{b^{3}} + \frac {c d}{b^{2}}\right ) + x \left (\frac {3 a^{2} d^{2}}{b^{4}} - \frac {4 a c d}{b^{3}} + \frac {c^{2}}{b^{2}}\right ) + \frac {- a^{4} d^{2} + 2 a^{3} b c d - a^{2} b^{2} c^{2}}{a b^{5} + b^{6} x} + \frac {d^{2} x^{3}}{3 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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