3.14.45 \(\int \frac {(a+b x)^3}{(c+d x)^2} \, dx\) [1345]

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

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^2,x]

[Out]

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

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Mathematica [A]
time = 0.02, size = 114, normalized size = 1.52 \begin {gather*} -\frac {b^2 (2 b c-3 a d) x}{d^3}+\frac {b^3 x^2}{2 d^2}+\frac {b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3}{d^4 (c+d x)}+\frac {3 \left (b^3 c^2-2 a b^2 c d+a^2 b d^2\right ) \log (c+d x)}{d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(c + d*x)^2,x]

[Out]

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

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Mathics [A]
time = 2.65, size = 107, normalized size = 1.43 \begin {gather*} \frac {-a^3 d^3+3 b \text {Log}\left [c+d x\right ] \left (c+d x\right ) \left (a d-b c\right )^2+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3+b^2 d x \left (3 a d-2 b c\right ) \left (c+d x\right )+\frac {b^3 d^2 x^2 \left (c+d x\right )}{2}}{d^4 \left (c+d x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 108, normalized size = 1.44

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{2} b d \,x^{2}+3 a d x -2 b c x \right )}{d^{3}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{d^{4} \left (d x +c \right )}+\frac {3 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4}}\)	\(108\)
norman	\(\frac {-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -3 b^{3} c^{3}}{d^{4}}+\frac {b^{3} x^{3}}{2 d}+\frac {3 b^{2} \left (2 a d -b c \right ) x^{2}}{2 d^{2}}}{d x +c}+\frac {3 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4}}\)	\(116\)
risch	\(\frac {b^{3} x^{2}}{2 d^{2}}+\frac {3 b^{2} a x}{d^{2}}-\frac {2 b^{3} c x}{d^{3}}-\frac {a^{3}}{d \left (d x +c \right )}+\frac {3 a^{2} b c}{d^{2} \left (d x +c \right )}-\frac {3 a \,b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {b^{3} c^{3}}{d^{4} \left (d x +c \right )}+\frac {3 b \ln \left (d x +c \right ) a^{2}}{d^{2}}-\frac {6 b^{2} \ln \left (d x +c \right ) a c}{d^{3}}+\frac {3 b^{3} \ln \left (d x +c \right ) c^{2}}{d^{4}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 117, normalized size = 1.56 \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}}{d^{5} x + c d^{4}} + \frac {b^{3} d x^{2} - 2 \, {\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x}{2 \, d^{3}} + \frac {3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (d x + c\right )}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (73) = 146\).
time = 0.29, size = 172, normalized size = 2.29 \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 (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} x + 6 \, {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{5} x + c d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [A]
time = 0.30, size = 102, normalized size = 1.36 \begin {gather*} \frac {b^{3} x^{2}}{2 d^{2}} + \frac {3 b \left (a d - b c\right )^{2} \log {\left (c + d x \right )}}{d^{4}} + x \left (\frac {3 a b^{2}}{d^{2}} - \frac {2 b^{3} c}{d^{3}}\right ) + \frac {- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}}{c d^{4} + d^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 126, normalized size = 1.68 \begin {gather*} \frac {\frac {1}{2} x^{2} b^{3} d^{2}-2 x b^{3} d c+3 x b^{2} a d^{2}}{d^{4}}+\frac {b^{3} c^{3}-3 b^{2} d c^{2} a+3 b d^{2} c a^{2}-d^{3} a^{3}}{d^{4} \left (x d+c\right )}+\frac {\left (3 b^{3} c^{2}-6 b^{2} a d c+3 b a^{2} d^{2}\right ) \ln \left |x d+c\right |}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.08, size = 123, normalized size = 1.64 \begin {gather*} x\,\left (\frac {3\,a\,b^2}{d^2}-\frac {2\,b^3\,c}{d^3}\right )+\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{d^4}-\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{d\,\left (x\,d^4+c\,d^3\right )}+\frac {b^3\,x^2}{2\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/(c + d*x)^2,x)

[Out]

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

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