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

Optimal. Leaf size=136 \[ \frac {a^3 (b c-a d)^2}{b^6 (a+b x)}+\frac {a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}-\frac {2 a x (b c-2 a d) (b c-a d)}{b^5}+\frac {x^2 (b c-3 a d) (b c-a d)}{2 b^4}+\frac {2 d x^3 (b c-a d)}{3 b^3}+\frac {d^2 x^4}{4 b^2} \]

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Rubi [A]  time = 0.13, antiderivative size = 136, 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^3 (b c-a d)^2}{b^6 (a+b x)}+\frac {a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}+\frac {2 d x^3 (b c-a d)}{3 b^3}+\frac {x^2 (b c-3 a d) (b c-a d)}{2 b^4}-\frac {2 a x (b c-2 a d) (b c-a d)}{b^5}+\frac {d^2 x^4}{4 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

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

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Mathematica [A]  time = 0.10, size = 149, normalized size = 1.10 \begin {gather*} \frac {\frac {12 a^3 (b c-a d)^2}{a+b x}+6 b^2 x^2 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-24 a b x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+12 a^2 \left (5 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (a+b x)+8 b^3 d x^3 (b c-a d)+3 b^4 d^2 x^4}{12 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a*b*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x + 6*b^2*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x^2 + 8*b^3*d*(b*c - a*
d)*x^3 + 3*b^4*d^2*x^4 + (12*a^3*(b*c - a*d)^2)/(a + b*x) + 12*a^2*(3*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*Log[a +
b*x])/(12*b^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (c+d x)^2}{(a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.92, size = 246, normalized size = 1.81 \begin {gather*} \frac {3 \, b^{5} d^{2} x^{5} + 12 \, a^{3} b^{2} c^{2} - 24 \, a^{4} b c d + 12 \, a^{5} d^{2} + {\left (8 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{4} + 2 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{4} c d + 5 \, a^{2} b^{3} d^{2}\right )} x^{3} - 6 \, {\left (3 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 5 \, a^{3} b^{2} d^{2}\right )} x^{2} - 24 \, {\left (a^{2} b^{3} c^{2} - 3 \, a^{3} b^{2} c d + 2 \, a^{4} b d^{2}\right )} x + 12 \, {\left (3 \, a^{3} b^{2} c^{2} - 8 \, a^{4} b c d + 5 \, a^{5} d^{2} + {\left (3 \, a^{2} b^{3} c^{2} - 8 \, a^{3} b^{2} c d + 5 \, a^{4} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^5*d^2*x^5 + 12*a^3*b^2*c^2 - 24*a^4*b*c*d + 12*a^5*d^2 + (8*b^5*c*d - 5*a*b^4*d^2)*x^4 + 2*(3*b^5*c^
2 - 8*a*b^4*c*d + 5*a^2*b^3*d^2)*x^3 - 6*(3*a*b^4*c^2 - 8*a^2*b^3*c*d + 5*a^3*b^2*d^2)*x^2 - 24*(a^2*b^3*c^2 -
3*a^3*b^2*c*d + 2*a^4*b*d^2)*x + 12*(3*a^3*b^2*c^2 - 8*a^4*b*c*d + 5*a^5*d^2 + (3*a^2*b^3*c^2 - 8*a^3*b^2*c*d
+ 5*a^4*b*d^2)*x)*log(b*x + a))/(b^7*x + a*b^6)

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giac [A]  time = 1.09, size = 236, normalized size = 1.74 \begin {gather*} \frac {{\left (3 \, d^{2} + \frac {4 \, {\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (b^{4} c^{2} - 8 \, a b^{3} c d + 10 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {12 \, {\left (3 \, a b^{5} c^{2} - 12 \, a^{2} b^{4} c d + 10 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{4}}{12 \, b^{6}} - \frac {{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} + \frac {\frac {a^{3} b^{6} c^{2}}{b x + a} - \frac {2 \, a^{4} b^{5} c d}{b x + a} + \frac {a^{5} b^{4} d^{2}}{b x + a}}{b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*d^2 + 4*(2*b^2*c*d - 5*a*b*d^2)/((b*x + a)*b) + 6*(b^4*c^2 - 8*a*b^3*c*d + 10*a^2*b^2*d^2)/((b*x + a)^
2*b^2) - 12*(3*a*b^5*c^2 - 12*a^2*b^4*c*d + 10*a^3*b^3*d^2)/((b*x + a)^3*b^3))*(b*x + a)^4/b^6 - (3*a^2*b^2*c^
2 - 8*a^3*b*c*d + 5*a^4*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^6 + (a^3*b^6*c^2/(b*x + a) - 2*a^4*b^5*c
*d/(b*x + a) + a^5*b^4*d^2/(b*x + a))/b^10

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maple [A]  time = 0.01, size = 205, normalized size = 1.51 \begin {gather*} \frac {d^{2} x^{4}}{4 b^{2}}-\frac {2 a \,d^{2} x^{3}}{3 b^{3}}+\frac {2 c d \,x^{3}}{3 b^{2}}+\frac {3 a^{2} d^{2} x^{2}}{2 b^{4}}-\frac {2 a c d \,x^{2}}{b^{3}}+\frac {c^{2} x^{2}}{2 b^{2}}+\frac {a^{5} d^{2}}{\left (b x +a \right ) b^{6}}-\frac {2 a^{4} c d}{\left (b x +a \right ) b^{5}}+\frac {5 a^{4} d^{2} \ln \left (b x +a \right )}{b^{6}}+\frac {a^{3} c^{2}}{\left (b x +a \right ) b^{4}}-\frac {8 a^{3} c d \ln \left (b x +a \right )}{b^{5}}-\frac {4 a^{3} d^{2} x}{b^{5}}+\frac {3 a^{2} c^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {6 a^{2} c d x}{b^{4}}-\frac {2 a \,c^{2} x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.02, size = 175, normalized size = 1.29 \begin {gather*} \frac {a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} d^{2} x^{4} + 8 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{2} - 24 \, {\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x}{12 \, b^{5}} + \frac {{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)/(b^7*x + a*b^6) + 1/12*(3*b^3*d^2*x^4 + 8*(b^3*c*d - a*b^2*d^2)*x^3 + 6*
(b^3*c^2 - 4*a*b^2*c*d + 3*a^2*b*d^2)*x^2 - 24*(a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x)/b^5 + (3*a^2*b^2*c^2 -
8*a^3*b*c*d + 5*a^4*d^2)*log(b*x + a)/b^6

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mupad [B]  time = 0.08, size = 236, normalized size = 1.74 \begin {gather*} x\,\left (\frac {a^2\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b^2}-\frac {2\,a\,\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 )}{b}\right )+x^2\,\left (\frac {c^2}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{2\,b^4}\right )-x^3\,\left (\frac {2\,a\,d^2}{3\,b^3}-\frac {2\,c\,d}{3\,b^2}\right )+\frac {a^5\,d^2-2\,a^4\,b\,c\,d+a^3\,b^2\,c^2}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,d^2-8\,a^3\,b\,c\,d+3\,a^2\,b^2\,c^2\right )}{b^6}+\frac {d^2\,x^4}{4\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(c + d*x)^2)/(a + b*x)^2,x)

[Out]

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

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sympy [A]  time = 0.65, size = 175, normalized size = 1.29 \begin {gather*} \frac {a^{2} \left (a d - b c\right ) \left (5 a d - 3 b c\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{3} \left (- \frac {2 a d^{2}}{3 b^{3}} + \frac {2 c d}{3 b^{2}}\right ) + x^{2} \left (\frac {3 a^{2} d^{2}}{2 b^{4}} - \frac {2 a c d}{b^{3}} + \frac {c^{2}}{2 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{2}}{b^{5}} + \frac {6 a^{2} c d}{b^{4}} - \frac {2 a c^{2}}{b^{3}}\right ) + \frac {a^{5} d^{2} - 2 a^{4} b c d + a^{3} b^{2} c^{2}}{a b^{6} + b^{7} x} + \frac {d^{2} x^{4}}{4 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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