3.12.25 \(\int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx\) [1125]

Optimal. Leaf size=99 \[ \frac {e (2 b B d+A b e-2 a B e) x}{b^3}+\frac {B e^2 x^2}{2 b^2}-\frac {(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) \log (a+b x)}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} -\frac {(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac {(b d-a e) \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^4}+\frac {e x (-2 a B e+A b e+2 b B d)}{b^3}+\frac {B e^2 x^2}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^2)/(a + b*x)^2,x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 153, normalized size = 1.55 \begin {gather*} \frac {e (2 b B d+A b e-2 a B e) x}{b^3}+\frac {B e^2 x^2}{2 b^2}+\frac {-A b^3 d^2+a b^2 B d^2+2 a A b^2 d e-2 a^2 b B d e-a^2 A b e^2+a^3 B e^2}{b^4 (a+b x)}+\frac {\left (b^2 B d^2+2 A b^2 d e-4 a b B d e-2 a A b e^2+3 a^2 B e^2\right ) \log (a+b x)}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^2)/(a + b*x)^2,x]

[Out]

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

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Maple [A]
time = 0.08, size = 151, normalized size = 1.53

method	result	size
default	\(\frac {e \left (\frac {1}{2} B b e \,x^{2}+A b e x -2 B a e x +2 B b d x \right )}{b^{3}}-\frac {A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e +A \,b^{3} d^{2}-B \,a^{3} e^{2}+2 B \,a^{2} b d e -B a \,b^{2} d^{2}}{b^{4} \left (b x +a \right )}+\frac {\left (-2 A a b \,e^{2}+2 A \,b^{2} d e +3 B \,a^{2} e^{2}-4 B a b d e +b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(151\)
norman	\(\frac {\frac {\left (2 A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e +A \,b^{3} d^{2}-3 B \,a^{3} e^{2}+4 B \,a^{2} b d e -B a \,b^{2} d^{2}\right ) x}{b^{3} a}+\frac {B \,e^{2} x^{3}}{2 b}+\frac {e \left (2 A b e -3 B a e +4 B b d \right ) x^{2}}{2 b^{2}}}{b x +a}-\frac {\left (2 A a b \,e^{2}-2 A \,b^{2} d e -3 B \,a^{2} e^{2}+4 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(165\)
risch	\(\frac {B \,e^{2} x^{2}}{2 b^{2}}+\frac {e^{2} A x}{b^{2}}-\frac {2 e^{2} B a x}{b^{3}}+\frac {2 e B d x}{b^{2}}-\frac {A \,a^{2} e^{2}}{b^{3} \left (b x +a \right )}+\frac {2 A a d e}{b^{2} \left (b x +a \right )}-\frac {A \,d^{2}}{b \left (b x +a \right )}+\frac {B \,a^{3} e^{2}}{b^{4} \left (b x +a \right )}-\frac {2 B \,a^{2} d e}{b^{3} \left (b x +a \right )}+\frac {B a \,d^{2}}{b^{2} \left (b x +a \right )}-\frac {2 \ln \left (b x +a \right ) A a \,e^{2}}{b^{3}}+\frac {2 \ln \left (b x +a \right ) A d e}{b^{2}}+\frac {3 \ln \left (b x +a \right ) B \,a^{2} e^{2}}{b^{4}}-\frac {4 \ln \left (b x +a \right ) B a d e}{b^{3}}+\frac {\ln \left (b x +a \right ) B \,d^{2}}{b^{2}}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 159, normalized size = 1.61 \begin {gather*} \frac {B a^{3} e^{2} - A a^{2} b e^{2} + {\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \, {\left (B a^{2} b e - A a b^{2} e\right )} d}{b^{5} x + a b^{4}} + \frac {B b x^{2} e^{2} + 2 \, {\left (2 \, B b d e - 2 \, B a e^{2} + A b e^{2}\right )} x}{2 \, b^{3}} + \frac {{\left (B b^{2} d^{2} + 3 \, B a^{2} e^{2} - 2 \, A a b e^{2} - 2 \, {\left (2 \, B a b e - A b^{2} e\right )} d\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*a^3*e^2 - A*a^2*b*e^2 + (B*a*b^2 - A*b^3)*d^2 - 2*(B*a^2*b*e - A*a*b^2*e)*d)/(b^5*x + a*b^4) + 1/2*(B*b*x^2
*e^2 + 2*(2*B*b*d*e - 2*B*a*e^2 + A*b*e^2)*x)/b^3 + (B*b^2*d^2 + 3*B*a^2*e^2 - 2*A*a*b*e^2 - 2*(2*B*a*b*e - A*
b^2*e)*d)*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. 235 vs. \(2 (102) = 204\).
time = 1.22, size = 235, normalized size = 2.37 \begin {gather*} \frac {2 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} + {\left (B b^{3} x^{3} + 2 \, B a^{3} - 2 \, A a^{2} b - {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} - 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x\right )} e^{2} + 4 \, {\left (B b^{3} d x^{2} + B a b^{2} d x - {\left (B a^{2} b - A a b^{2}\right )} d\right )} e + 2 \, {\left (B b^{3} d^{2} x + B a b^{2} d^{2} + {\left (3 \, B a^{3} - 2 \, A a^{2} b + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} e^{2} - 2 \, {\left ({\left (2 \, B a b^{2} - A b^{3}\right )} d x + {\left (2 \, B a^{2} b - A a b^{2}\right )} d\right )} e\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((B*x+A)*(e*x+d)^2/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(2*(B*a*b^2 - A*b^3)*d^2 + (B*b^3*x^3 + 2*B*a^3 - 2*A*a^2*b - (3*B*a*b^2 - 2*A*b^3)*x^2 - 2*(2*B*a^2*b - A
*a*b^2)*x)*e^2 + 4*(B*b^3*d*x^2 + B*a*b^2*d*x - (B*a^2*b - A*a*b^2)*d)*e + 2*(B*b^3*d^2*x + B*a*b^2*d^2 + (3*B
*a^3 - 2*A*a^2*b + (3*B*a^2*b - 2*A*a*b^2)*x)*e^2 - 2*((2*B*a*b^2 - A*b^3)*d*x + (2*B*a^2*b - A*a*b^2)*d)*e)*l
og(b*x + a))/(b^5*x + a*b^4)

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Sympy [A]
time = 0.55, size = 151, normalized size = 1.53 \begin {gather*} \frac {B e^{2} x^{2}}{2 b^{2}} + x \left (\frac {A e^{2}}{b^{2}} - \frac {2 B a e^{2}}{b^{3}} + \frac {2 B d e}{b^{2}}\right ) + \frac {- A a^{2} b e^{2} + 2 A a b^{2} d e - A b^{3} d^{2} + B a^{3} e^{2} - 2 B a^{2} b d e + B a b^{2} d^{2}}{a b^{4} + b^{5} x} + \frac {\left (a e - b d\right ) \left (- 2 A b e + 3 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**2/(b*x+a)**2,x)

[Out]

B*e**2*x**2/(2*b**2) + x*(A*e**2/b**2 - 2*B*a*e**2/b**3 + 2*B*d*e/b**2) + (-A*a**2*b*e**2 + 2*A*a*b**2*d*e - A
*b**3*d**2 + B*a**3*e**2 - 2*B*a**2*b*d*e + B*a*b**2*d**2)/(a*b**4 + b**5*x) + (a*e - b*d)*(-2*A*b*e + 3*B*a*e
- B*b*d)*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. 227 vs. \(2 (102) = 204\).
time = 1.53, size = 227, normalized size = 2.29 \begin {gather*} \frac {{\left (b x + a\right )}^{2} {\left (B e^{2} + \frac {2 \, {\left (2 \, B b^{2} d e - 3 \, B a b e^{2} + A b^{2} e^{2}\right )}}{{\left (b x + a\right )} b}\right )}}{2 \, b^{4}} - \frac {{\left (B b^{2} d^{2} - 4 \, B a b d e + 2 \, A b^{2} d e + 3 \, B a^{2} e^{2} - 2 \, A a b e^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {\frac {B a b^{4} d^{2}}{b x + a} - \frac {A b^{5} d^{2}}{b x + a} - \frac {2 \, B a^{2} b^{3} d e}{b x + a} + \frac {2 \, A a b^{4} d e}{b x + a} + \frac {B a^{3} b^{2} e^{2}}{b x + a} - \frac {A a^{2} b^{3} e^{2}}{b x + a}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2/(b*x+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.11, size = 165, normalized size = 1.67 \begin {gather*} x\,\left (\frac {A\,e^2+2\,B\,d\,e}{b^2}-\frac {2\,B\,a\,e^2}{b^3}\right )+\frac {\ln \left (a+b\,x\right )\,\left (3\,B\,a^2\,e^2-4\,B\,a\,b\,d\,e-2\,A\,a\,b\,e^2+B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{b^4}-\frac {-B\,a^3\,e^2+2\,B\,a^2\,b\,d\,e+A\,a^2\,b\,e^2-B\,a\,b^2\,d^2-2\,A\,a\,b^2\,d\,e+A\,b^3\,d^2}{b\,\left (x\,b^4+a\,b^3\right )}+\frac {B\,e^2\,x^2}{2\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^2)/(a + b*x)^2,x)

[Out]

x*((A*e^2 + 2*B*d*e)/b^2 - (2*B*a*e^2)/b^3) + (log(a + b*x)*(3*B*a^2*e^2 + B*b^2*d^2 - 2*A*a*b*e^2 + 2*A*b^2*d
*e - 4*B*a*b*d*e))/b^4 - (A*b^3*d^2 - B*a^3*e^2 + A*a^2*b*e^2 - B*a*b^2*d^2 - 2*A*a*b^2*d*e + 2*B*a^2*b*d*e)/(
b*(a*b^3 + b^4*x)) + (B*e^2*x^2)/(2*b^2)

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