3.13.54 \(\int \frac {(c+d x)^2}{(a+b x)^4} \, dx\) [1254]

Optimal. Leaf size=28 \[ -\frac {(c+d x)^3}{3 (b c-a d) (a+b x)^3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \begin {gather*} -\frac {(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*x)^4,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 53, normalized size = 1.89 \begin {gather*} -\frac {a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )}{3 b^3 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*x)^4,x]

[Out]

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

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(28)=56\).
time = 2.39, size = 80, normalized size = 2.86 \begin {gather*} \frac {-a^2 d^2-a b c d-b^2 c^2-3 b d x \left (a d+b c\right )-3 b^2 d^2 x^2}{3 b^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(c + d*x)^2/(a + b*x)^4,x]')

[Out]

(-a ^ 2 d ^ 2 - a b c d - b ^ 2 c ^ 2 - 3 b d x (a d + b c) - 3 b ^ 2 d ^ 2 x ^ 2) / (3 b ^ 3 (a ^ 3 + 3 a ^ 2
b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(26)=52\).
time = 0.14, size = 70, normalized size = 2.50

method	result	size
gosper	\(-\frac {3 d^{2} x^{2} b^{2}+3 a b \,d^{2} x +3 b^{2} c d x +a^{2} d^{2}+a b c d +b^{2} c^{2}}{3 \left (b x +a \right )^{3} b^{3}}\)	\(60\)
risch	\(\frac {-\frac {d^{2} x^{2}}{b}-\frac {d \left (a d +b c \right ) x}{b^{2}}-\frac {a^{2} d^{2}+a b c d +b^{2} c^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(60\)
norman	\(\frac {-\frac {d^{2} x^{2}}{b}+\frac {\left (-a \,d^{2}-b d c \right ) x}{b^{2}}+\frac {-a^{2} d^{2}-a b c d -b^{2} c^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(66\)
default	\(-\frac {d^{2}}{b^{3} \left (b x +a \right )}+\frac {d \left (a d -b c \right )}{b^{3} \left (b x +a \right )^{2}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
time = 0.29, size = 84, normalized size = 3.00 \begin {gather*} -\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
time = 0.30, size = 84, normalized size = 3.00 \begin {gather*} -\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
time = 0.35, size = 88, normalized size = 3.14 \begin {gather*} \frac {- a^{2} d^{2} - a b c d - b^{2} c^{2} - 3 b^{2} d^{2} x^{2} + x \left (- 3 a b d^{2} - 3 b^{2} c d\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(b*x+a)**4,x)

[Out]

(-a**2*d**2 - a*b*c*d - b**2*c**2 - 3*b**2*d**2*x**2 + x*(-3*a*b*d**2 - 3*b**2*c*d))/(3*a**3*b**3 + 9*a**2*b**
4*x + 9*a*b**5*x**2 + 3*b**6*x**3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
time = 0.00, size = 67, normalized size = 2.39 \begin {gather*} \frac {-3 x^{2} d^{2} b^{2}-3 x d^{2} b a-3 x d c b^{2}-d^{2} a^{2}-d c b a-c^{2} b^{2}}{3 b^{3} \left (x b+a\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^4,x)

[Out]

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

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Mupad [B]
time = 0.04, size = 80, normalized size = 2.86 \begin {gather*} -\frac {\frac {a^2\,d^2+a\,b\,c\,d+b^2\,c^2}{3\,b^3}+\frac {d^2\,x^2}{b}+\frac {d\,x\,\left (a\,d+b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(a + b*x)^4,x)

[Out]

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

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