3.3.73 \(\int \frac {(c+d x)^3}{(a+b x)^2} \, dx\) [273]

Optimal. Leaf size=75 \[ \frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^2}{2 b^2}-\frac {(b c-a d)^3}{b^4 (a+b x)}+\frac {3 d (b c-a d)^2 \log (a+b x)}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} -\frac {(b c-a d)^3}{b^4 (a+b x)}+\frac {3 d (b c-a d)^2 \log (a+b x)}{b^4}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {d^3 x^2}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*x)^2,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 72, normalized size = 0.96 \begin {gather*} \frac {2 b d^2 (3 b c-2 a d) x+b^2 d^3 x^2-\frac {2 (b c-a d)^3}{a+b x}+6 d (b c-a d)^2 \log (a+b x)}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + b*x)^2,x]

[Out]

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

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Maple [A]
time = 0.06, size = 109, normalized size = 1.45

method	result	size
default	\(-\frac {d^{2} \left (-\frac {1}{2} b d \,x^{2}+2 a d x -3 b c x \right )}{b^{3}}+\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{b^{4} \left (b x +a \right )}\)	\(109\)
norman	\(\frac {\frac {3 a^{3} d^{3}-6 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{b^{4}}+\frac {d^{3} x^{3}}{2 b}-\frac {3 d^{2} \left (a d -2 b c \right ) x^{2}}{2 b^{2}}}{b x +a}+\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(115\)
risch	\(\frac {d^{3} x^{2}}{2 b^{2}}-\frac {2 d^{3} a x}{b^{3}}+\frac {3 c \,d^{2} x}{b^{2}}+\frac {a^{3} d^{3}}{b^{4} \left (b x +a \right )}-\frac {3 a^{2} c \,d^{2}}{b^{3} \left (b x +a \right )}+\frac {3 a \,c^{2} d}{b^{2} \left (b x +a \right )}-\frac {c^{3}}{b \left (b x +a \right )}+\frac {3 d^{3} \ln \left (b x +a \right ) a^{2}}{b^{4}}-\frac {6 d^{2} \ln \left (b x +a \right ) a c}{b^{3}}+\frac {3 d \ln \left (b x +a \right ) c^{2}}{b^{2}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 118, normalized size = 1.57 \begin {gather*} -\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{b^{5} x + a b^{4}} + \frac {b d^{3} x^{2} + 2 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{2 \, b^{3}} + \frac {3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (73) = 146\).
time = 0.63, size = 173, normalized size = 2.31 \begin {gather*} \frac {b^{3} d^{3} x^{3} - 2 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + 3 \, {\left (2 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x + 6 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.27, size = 102, normalized size = 1.36 \begin {gather*} x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{a b^{4} + b^{5} x} + \frac {d^{3} x^{2}}{2 b^{2}} + \frac {3 d \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (73) = 146\).
time = 2.66, size = 167, normalized size = 2.23 \begin {gather*} \frac {{\left (d^{3} + \frac {6 \, {\left (b^{2} c d^{2} - a b d^{3}\right )}}{{\left (b x + a\right )} b}\right )} {\left (b x + a\right )}^{2}}{2 \, b^{4}} - \frac {3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} - \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.38, size = 123, normalized size = 1.64 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (3\,a^2\,d^3-6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{b^4}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )+\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{b\,\left (x\,b^4+a\,b^3\right )}+\frac {d^3\,x^2}{2\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + b*x)^2,x)

[Out]

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

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