3.14.46 \(\int \frac {(a+b x)^2}{(c+d x)^2} \, dx\) [1346]

Optimal. Leaf size=51 \[ \frac {b^2 x}{d^2}-\frac {(b c-a d)^2}{d^3 (c+d x)}-\frac {2 b (b c-a d) \log (c+d x)}{d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 51, 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)^2}{d^3 (c+d x)}-\frac {2 b (b c-a d) \log (c+d x)}{d^3}+\frac {b^2 x}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(c + d*x)^2,x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 47, normalized size = 0.92 \begin {gather*} \frac {b^2 d x-\frac {(b c-a d)^2}{c+d x}+2 b (-b c+a d) \log (c+d x)}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(c + d*x)^2,x]

[Out]

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

________________________________________________________________________________________

Mathics [A]
time = 2.28, size = 67, normalized size = 1.31 \begin {gather*} \frac {-a^2 d^2+2 b \text {Log}\left [c+d x\right ] \left (a d-b c\right ) \left (c+d x\right )+2 a b c d-b^2 c^2+b^2 d x \left (c+d x\right )}{d^3 \left (c+d x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 63, normalized size = 1.24

method	result	size
default	\(\frac {b^{2} x}{d^{2}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {2 b \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{3}}\)	\(63\)
norman	\(\frac {\frac {b^{2} x^{2}}{d}-\frac {a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}}{d^{3}}}{d x +c}+\frac {2 b \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{3}}\)	\(68\)
risch	\(\frac {b^{2} x}{d^{2}}-\frac {a^{2}}{d \left (d x +c \right )}+\frac {2 a b c}{d^{2} \left (d x +c \right )}-\frac {b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {2 b \ln \left (d x +c \right ) a}{d^{2}}-\frac {2 b^{2} \ln \left (d x +c \right ) c}{d^{3}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 67, normalized size = 1.31 \begin {gather*} \frac {b^{2} x}{d^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{4} x + c d^{3}} - \frac {2 \, {\left (b^{2} c - a b d\right )} \log \left (d x + c\right )}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 92, normalized size = 1.80 \begin {gather*} \frac {b^{2} d^{2} x^{2} + b^{2} c d x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{d^{4} x + c d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.20, size = 60, normalized size = 1.18 \begin {gather*} \frac {b^{2} x}{d^{2}} + \frac {2 b \left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{3}} + \frac {- a^{2} d^{2} + 2 a b c d - b^{2} c^{2}}{c d^{3} + d^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 69, normalized size = 1.35 \begin {gather*} \frac {x b^{2}}{d^{2}}+\frac {-b^{2} c^{2}+2 b d c a-d^{2} a^{2}}{d^{3} \left (x d+c\right )}+\frac {\left (-2 b^{2} c+2 b a d\right ) \ln \left |x d+c\right |}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.24, size = 71, normalized size = 1.39 \begin {gather*} \frac {b^2\,x}{d^2}-\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{d\,\left (x\,d^3+c\,d^2\right )}-\frac {\ln \left (c+d\,x\right )\,\left (2\,b^2\,c-2\,a\,b\,d\right )}{d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(c + d*x)^2,x)

[Out]

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

________________________________________________________________________________________