3.15.48 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=101 \[ -\frac {(a+b x)^3 (B d-A e)}{3 e (d+e x)^3 (b d-a e)}+\frac {2 b B (b d-a e)}{e^4 (d+e x)}-\frac {B (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac {b^2 B \log (d+e x)}{e^4} \]

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Rubi [A]  time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {27, 78, 43} \begin {gather*} -\frac {(a+b x)^3 (B d-A e)}{3 e (d+e x)^3 (b d-a e)}+\frac {2 b B (b d-a e)}{e^4 (d+e x)}-\frac {B (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac {b^2 B \log (d+e x)}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^4,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

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])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 138, normalized size = 1.37 \begin {gather*} \frac {-a^2 e^2 (2 A e+B (d+3 e x))-2 a b e \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+b^2 \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 b^2 B (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^4,x]

[Out]

(-(a^2*e^2*(2*A*e + B*(d + 3*e*x))) - 2*a*b*e*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)) + b^2*(-2*A*
e*(d^2 + 3*d*e*x + 3*e^2*x^2) + B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + 6*b^2*B*(d + e*x)^3*Log[d + e*x])/(6*e
^4*(d + e*x)^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^4, x]

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fricas [B]  time = 0.41, size = 221, normalized size = 2.19 \begin {gather*} \frac {11 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e - {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 3 \, {\left (9 \, B b^{2} d^{2} e - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} - {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x + 6 \, {\left (B b^{2} e^{3} x^{3} + 3 \, B b^{2} d e^{2} x^{2} + 3 \, B b^{2} d^{2} e x + B b^{2} d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(11*B*b^2*d^3 - 2*A*a^2*e^3 - 2*(2*B*a*b + A*b^2)*d^2*e - (B*a^2 + 2*A*a*b)*d*e^2 + 6*(3*B*b^2*d*e^2 - (2*
B*a*b + A*b^2)*e^3)*x^2 + 3*(9*B*b^2*d^2*e - 2*(2*B*a*b + A*b^2)*d*e^2 - (B*a^2 + 2*A*a*b)*e^3)*x + 6*(B*b^2*e
^3*x^3 + 3*B*b^2*d*e^2*x^2 + 3*B*b^2*d^2*e*x + B*b^2*d^3)*log(e*x + d))/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x +
d^3*e^4)

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giac [A]  time = 0.16, size = 163, normalized size = 1.61 \begin {gather*} B b^{2} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (6 \, {\left (3 \, B b^{2} d e - 2 \, B a b e^{2} - A b^{2} e^{2}\right )} x^{2} + 3 \, {\left (9 \, B b^{2} d^{2} - 4 \, B a b d e - 2 \, A b^{2} d e - B a^{2} e^{2} - 2 \, A a b e^{2}\right )} x + {\left (11 \, B b^{2} d^{3} - 4 \, B a b d^{2} e - 2 \, A b^{2} d^{2} e - B a^{2} d e^{2} - 2 \, A a b d e^{2} - 2 \, A a^{2} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x, algorithm="giac")

[Out]

B*b^2*e^(-4)*log(abs(x*e + d)) + 1/6*(6*(3*B*b^2*d*e - 2*B*a*b*e^2 - A*b^2*e^2)*x^2 + 3*(9*B*b^2*d^2 - 4*B*a*b
*d*e - 2*A*b^2*d*e - B*a^2*e^2 - 2*A*a*b*e^2)*x + (11*B*b^2*d^3 - 4*B*a*b*d^2*e - 2*A*b^2*d^2*e - B*a^2*d*e^2
- 2*A*a*b*d*e^2 - 2*A*a^2*e^3)*e^(-1))*e^(-3)/(x*e + d)^3

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maple [B]  time = 0.06, size = 251, normalized size = 2.49 \begin {gather*} -\frac {A \,a^{2}}{3 \left (e x +d \right )^{3} e}+\frac {2 A a b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {A \,b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {B \,a^{2} d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 B a b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {B \,b^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A a b}{\left (e x +d \right )^{2} e^{2}}+\frac {A \,b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {B \,a^{2}}{2 \left (e x +d \right )^{2} e^{2}}+\frac {2 B a b d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 B \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {A \,b^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 B a b}{\left (e x +d \right ) e^{3}}+\frac {3 B \,b^{2} d}{\left (e x +d \right ) e^{4}}+\frac {B \,b^{2} \ln \left (e x +d \right )}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x)

[Out]

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

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maxima [A]  time = 0.53, size = 184, normalized size = 1.82 \begin {gather*} \frac {11 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e - {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 3 \, {\left (9 \, B b^{2} d^{2} e - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} - {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {B b^{2} \log \left (e x + d\right )}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(11*B*b^2*d^3 - 2*A*a^2*e^3 - 2*(2*B*a*b + A*b^2)*d^2*e - (B*a^2 + 2*A*a*b)*d*e^2 + 6*(3*B*b^2*d*e^2 - (2*
B*a*b + A*b^2)*e^3)*x^2 + 3*(9*B*b^2*d^2*e - 2*(2*B*a*b + A*b^2)*d*e^2 - (B*a^2 + 2*A*a*b)*e^3)*x)/(e^7*x^3 +
3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + B*b^2*log(e*x + d)/e^4

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mupad [B]  time = 0.12, size = 178, normalized size = 1.76 \begin {gather*} \frac {B\,b^2\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {B\,a^2\,d\,e^2+2\,A\,a^2\,e^3+4\,B\,a\,b\,d^2\,e+2\,A\,a\,b\,d\,e^2-11\,B\,b^2\,d^3+2\,A\,b^2\,d^2\,e}{6\,e^4}+\frac {x\,\left (B\,a^2\,e^2+4\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2-9\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{2\,e^3}+\frac {b\,x^2\,\left (A\,b\,e+2\,B\,a\,e-3\,B\,b\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x))/(d + e*x)^4,x)

[Out]

(B*b^2*log(d + e*x))/e^4 - ((2*A*a^2*e^3 - 11*B*b^2*d^3 + 2*A*b^2*d^2*e + B*a^2*d*e^2 + 2*A*a*b*d*e^2 + 4*B*a*
b*d^2*e)/(6*e^4) + (x*(B*a^2*e^2 - 9*B*b^2*d^2 + 2*A*a*b*e^2 + 2*A*b^2*d*e + 4*B*a*b*d*e))/(2*e^3) + (b*x^2*(A
*b*e + 2*B*a*e - 3*B*b*d))/e^2)/(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)

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sympy [B]  time = 5.76, size = 211, normalized size = 2.09 \begin {gather*} \frac {B b^{2} \log {\left (d + e x \right )}}{e^{4}} + \frac {- 2 A a^{2} e^{3} - 2 A a b d e^{2} - 2 A b^{2} d^{2} e - B a^{2} d e^{2} - 4 B a b d^{2} e + 11 B b^{2} d^{3} + x^{2} \left (- 6 A b^{2} e^{3} - 12 B a b e^{3} + 18 B b^{2} d e^{2}\right ) + x \left (- 6 A a b e^{3} - 6 A b^{2} d e^{2} - 3 B a^{2} e^{3} - 12 B a b d e^{2} + 27 B b^{2} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**4,x)

[Out]

B*b**2*log(d + e*x)/e**4 + (-2*A*a**2*e**3 - 2*A*a*b*d*e**2 - 2*A*b**2*d**2*e - B*a**2*d*e**2 - 4*B*a*b*d**2*e
+ 11*B*b**2*d**3 + x**2*(-6*A*b**2*e**3 - 12*B*a*b*e**3 + 18*B*b**2*d*e**2) + x*(-6*A*a*b*e**3 - 6*A*b**2*d*e
**2 - 3*B*a**2*e**3 - 12*B*a*b*d*e**2 + 27*B*b**2*d**2*e))/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*
e**7*x**3)

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