3.3.14 \(\int \frac {x^2 (c+d x)^2}{a+b x} \, dx\) [214]

Optimal. Leaf size=94 \[ -\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^2}{2 b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^4}{4 b}+\frac {a^2 (b c-a d)^2 \log (a+b x)}{b^5} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 94, 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 = {90} \begin {gather*} \frac {a^2 (b c-a d)^2 \log (a+b x)}{b^5}-\frac {a x (b c-a d)^2}{b^4}+\frac {x^2 (b c-a d)^2}{2 b^3}+\frac {d x^3 (2 b c-a d)}{3 b^2}+\frac {d^2 x^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^2)/(a + b*x),x]

[Out]

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

Rule 90

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

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Mathematica [A]
time = 0.03, size = 103, normalized size = 1.10 \begin {gather*} \frac {b x \left (-12 a^3 d^2+6 a^2 b d (4 c+d x)-4 a b^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^3 x \left (6 c^2+8 c d x+3 d^2 x^2\right )\right )+12 a^2 (b c-a d)^2 \log (a+b x)}{12 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.08, size = 134, normalized size = 1.43

method	result	size
norman	\(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 b^{3}}+\frac {d^{2} x^{4}}{4 b}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{b^{4}}-\frac {d \left (a d -2 b c \right ) x^{3}}{3 b^{2}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(121\)
default	\(-\frac {-\frac {d^{2} x^{4} b^{3}}{4}+\frac {\left (\left (a d -b c \right ) b^{2} d -b^{3} d c \right ) x^{3}}{3}+\frac {\left (\left (a d -b c \right ) b^{2} c -b d \left (a^{2} d -a b c \right )\right ) x^{2}}{2}+\left (a d -b c \right ) \left (a^{2} d -a b c \right ) x}{b^{4}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(134\)
risch	\(\frac {d^{2} x^{4}}{4 b}-\frac {x^{3} a \,d^{2}}{3 b^{2}}+\frac {2 x^{3} d c}{3 b}-\frac {x^{2} a c d}{b^{2}}+\frac {x^{2} c^{2}}{2 b}+\frac {x^{2} a^{2} d^{2}}{2 b^{3}}-\frac {a^{3} d^{2} x}{b^{4}}+\frac {2 a^{2} c d x}{b^{3}}-\frac {a \,c^{2} x}{b^{2}}+\frac {a^{4} \ln \left (b x +a \right ) d^{2}}{b^{5}}-\frac {2 a^{3} \ln \left (b x +a \right ) c d}{b^{4}}+\frac {a^{2} \ln \left (b x +a \right ) c^{2}}{b^{3}}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.37, size = 132, normalized size = 1.40 \begin {gather*} \frac {3 \, b^{3} d^{2} x^{4} + 4 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2} - 12 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{12 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.52, size = 134, normalized size = 1.43 \begin {gather*} \frac {3 \, b^{4} d^{2} x^{4} + 4 \, {\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} - 12 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x + 12 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x + a\right )}{12 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.15, size = 116, normalized size = 1.23 \begin {gather*} \frac {a^{2} \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{5}} + x^{3} \left (- \frac {a d^{2}}{3 b^{2}} + \frac {2 c d}{3 b}\right ) + x^{2} \left (\frac {a^{2} d^{2}}{2 b^{3}} - \frac {a c d}{b^{2}} + \frac {c^{2}}{2 b}\right ) + x \left (- \frac {a^{3} d^{2}}{b^{4}} + \frac {2 a^{2} c d}{b^{3}} - \frac {a c^{2}}{b^{2}}\right ) + \frac {d^{2} x^{4}}{4 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.30, size = 139, normalized size = 1.48 \begin {gather*} \frac {3 \, b^{3} d^{2} x^{4} + 8 \, b^{3} c d x^{3} - 4 \, a b^{2} d^{2} x^{3} + 6 \, b^{3} c^{2} x^{2} - 12 \, a b^{2} c d x^{2} + 6 \, a^{2} b d^{2} x^{2} - 12 \, a b^{2} c^{2} x + 24 \, a^{2} b c d x - 12 \, a^{3} d^{2} x}{12 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left ({\left | b x + a \right |}\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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 141, normalized size = 1.50 \begin {gather*} x^2\,\left (\frac {c^2}{2\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{2\,b}\right )-x^3\,\left (\frac {a\,d^2}{3\,b^2}-\frac {2\,c\,d}{3\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2\right )}{b^5}+\frac {d^2\,x^4}{4\,b}-\frac {a\,x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x)^2)/(a + b*x),x)

[Out]

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

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