3.13.55 \(\int \frac {(c+d x)^2}{(a+b x)^5} \, dx\) [1255]

Optimal. Leaf size=65 \[ -\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {d^2}{2 b^3 (a+b x)^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 65, 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 {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {d^2}{2 b^3 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*x)^5,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 56, normalized size = 0.86 \begin {gather*} -\frac {a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )}{12 b^3 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*x)^5,x]

[Out]

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

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Mathics [A]
time = 2.65, size = 92, normalized size = 1.42 \begin {gather*} \frac {-a^2 d^2-2 a b c d-3 b^2 c^2-4 b d x \left (a d+2 b c\right )-6 b^2 d^2 x^2}{12 b^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(c + d*x)^2/(a + b*x)^5,x]')

[Out]

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

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Maple [A]
time = 0.14, size = 71, normalized size = 1.09

method	result	size
gosper	\(-\frac {6 d^{2} x^{2} b^{2}+4 a b \,d^{2} x +8 b^{2} c d x +a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{12 b^{3} \left (b x +a \right )^{4}}\)	\(62\)
risch	\(\frac {-\frac {d^{2} x^{2}}{2 b}-\frac {d \left (a d +2 b c \right ) x}{3 b^{2}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{12 b^{3}}}{\left (b x +a \right )^{4}}\)	\(63\)
default	\(-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{4 b^{3} \left (b x +a \right )^{4}}-\frac {d^{2}}{2 b^{3} \left (b x +a \right )^{2}}+\frac {2 d \left (a d -b c \right )}{3 b^{3} \left (b x +a \right )^{3}}\)	\(71\)
norman	\(\frac {-\frac {d^{2} x^{2}}{2 b}+\frac {\left (-a b \,d^{2}-2 b^{2} c d \right ) x}{3 b^{3}}+\frac {-a^{2} b \,d^{2}-2 a \,b^{2} c d -3 b^{3} c^{2}}{12 b^{4}}}{\left (b x +a \right )^{4}}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 98, normalized size = 1.51 \begin {gather*} -\frac {6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.29, size = 98, normalized size = 1.51 \begin {gather*} -\frac {6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.44, size = 104, normalized size = 1.60 \begin {gather*} \frac {- a^{2} d^{2} - 2 a b c d - 3 b^{2} c^{2} - 6 b^{2} d^{2} x^{2} + x \left (- 4 a b d^{2} - 8 b^{2} c d\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(b*x+a)**5,x)

[Out]

(-a**2*d**2 - 2*a*b*c*d - 3*b**2*c**2 - 6*b**2*d**2*x**2 + x*(-4*a*b*d**2 - 8*b**2*c*d))/(12*a**4*b**3 + 48*a*
*3*b**4*x + 72*a**2*b**5*x**2 + 48*a*b**6*x**3 + 12*b**7*x**4)

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Giac [A]
time = 0.00, size = 69, normalized size = 1.06 \begin {gather*} \frac {-6 x^{2} d^{2} b^{2}-4 x d^{2} b a-8 x d c b^{2}-d^{2} a^{2}-2 d c b a-3 c^{2} b^{2}}{12 b^{3} \left (x b+a\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(b*x+a)^5,x)

[Out]

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

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Mupad [B]
time = 0.19, size = 39, normalized size = 0.60 \begin {gather*} \frac {{\left (c+d\,x\right )}^3\,\left (4\,a\,d-3\,b\,c+b\,d\,x\right )}{12\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(a + b*x)^5,x)

[Out]

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

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