3.12.30 \(\int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx\) [1130]

Optimal. Leaf size=157 \[ -\frac {b (A b-a B)}{(b d-a e)^3 (a+b x)}+\frac {B d-A e}{2 (b d-a e)^2 (d+e x)^2}+\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (d+e x)}+\frac {b (b B d-3 A b e+2 a B e) \log (a+b x)}{(b d-a e)^4}-\frac {b (b B d-3 A b e+2 a B e) \log (d+e x)}{(b d-a e)^4} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 157, 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 {b (A b-a B)}{(a+b x) (b d-a e)^3}+\frac {a B e-2 A b e+b B d}{(d+e x) (b d-a e)^3}+\frac {B d-A e}{2 (d+e x)^2 (b d-a e)^2}+\frac {b \log (a+b x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4}-\frac {b \log (d+e x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^2*(d + e*x)^3),x]

[Out]

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

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Mathematica [A]
time = 0.10, size = 146, normalized size = 0.93 \begin {gather*} \frac {-\frac {2 b (A b-a B) (b d-a e)}{a+b x}+\frac {(b d-a e)^2 (B d-A e)}{(d+e x)^2}+\frac {2 (b d-a e) (b B d-2 A b e+a B e)}{d+e x}+2 b (b B d-3 A b e+2 a B e) \log (a+b x)-2 b (b B d-3 A b e+2 a B e) \log (d+e x)}{2 (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^3),x]

[Out]

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

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Maple [A]
time = 0.11, size = 159, normalized size = 1.01

method	result	size
default	\(-\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {\left (A b -B a \right ) b}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {A e -B d}{2 \left (a e -b d \right )^{2} \left (e x +d \right )^{2}}+\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {2 A b e -B a e -B b d}{\left (a e -b d \right )^{3} \left (e x +d \right )}\)	\(159\)
norman	\(\frac {\frac {\left (3 A \,b^{3} e^{3}-2 B a \,b^{2} e^{3}-b^{3} B d \,e^{2}\right ) x^{2}}{e b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {A \,a^{2} b \,e^{4}-5 A a \,b^{2} d \,e^{3}-2 A \,b^{3} d^{2} e^{2}+B \,a^{2} b d \,e^{3}+5 B a \,b^{2} d^{2} e^{2}}{2 b \,e^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (3 A a \,b^{2} e^{4}+9 A \,b^{3} d \,e^{3}-2 B \,a^{2} b \,e^{4}-7 B a \,b^{2} d \,e^{3}-3 b^{3} B \,d^{2} e^{2}\right ) x}{2 b \,e^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (b x +a \right ) \left (e x +d \right )^{2}}+\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (e x +d \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (b x +a \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\)	\(450\)
risch	\(\frac {\frac {b e \left (3 A b e -2 B a e -B b d \right ) x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\left (a e +3 b d \right ) \left (3 A b e -2 B a e -B b d \right ) x}{2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}}-\frac {a^{2} A \,e^{2}-5 A a b d e -2 A \,b^{2} d^{2}+B \,a^{2} d e +5 B a b \,d^{2}}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (b x +a \right ) \left (e x +d \right )^{2}}-\frac {3 b^{2} \ln \left (b x +a \right ) A e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {2 b \ln \left (b x +a \right ) B a e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {b^{2} \ln \left (b x +a \right ) B d}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {3 b^{2} \ln \left (-e x -d \right ) A e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {2 b \ln \left (-e x -d \right ) B a e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {b^{2} \ln \left (-e x -d \right ) B d}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\)	\(613\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^2/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (169) = 338\).
time = 0.31, size = 472, normalized size = 3.01 \begin {gather*} \frac {{\left (B b^{2} d + 2 \, B a b e - 3 \, A b^{2} e\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {{\left (B b^{2} d + 2 \, B a b e - 3 \, A b^{2} e\right )} \log \left (x e + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {A a^{2} e^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} d^{2} + 2 \, {\left (B b^{2} d e + 2 \, B a b e^{2} - 3 \, A b^{2} e^{2}\right )} x^{2} + {\left (B a^{2} e - 5 \, A a b e\right )} d + {\left (3 \, B b^{2} d^{2} + 2 \, B a^{2} e^{2} - 3 \, A a b e^{2} + {\left (7 \, B a b e - 9 \, A b^{2} e\right )} d\right )} x}{2 \, {\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} + {\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} + {\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} + {\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (169) = 338\).
time = 0.92, size = 792, normalized size = 5.04 \begin {gather*} \frac {3 \, B b^{3} d^{3} x + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} d^{3} - {\left (A a^{3} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x\right )} e^{3} + {\left (2 \, {\left (B a b^{2} - 3 \, A b^{3}\right )} d x^{2} - {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} d x - {\left (B a^{3} - 6 \, A a^{2} b\right )} d\right )} e^{2} + {\left (2 \, B b^{3} d^{2} x^{2} + {\left (4 \, B a b^{2} - 9 \, A b^{3}\right )} d^{2} x - {\left (4 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{2}\right )} e + 2 \, {\left (B b^{3} d^{3} x + B a b^{2} d^{3} + {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} e^{3} + {\left (B b^{3} d x^{3} + {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} d x^{2} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d x\right )} e^{2} + {\left (2 \, B b^{3} d^{2} x^{2} + {\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} d^{2} x + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2}\right )} e\right )} \log \left (b x + a\right ) - 2 \, {\left (B b^{3} d^{3} x + B a b^{2} d^{3} + {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} e^{3} + {\left (B b^{3} d x^{3} + {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} d x^{2} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d x\right )} e^{2} + {\left (2 \, B b^{3} d^{2} x^{2} + {\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} d^{2} x + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2}\right )} e\right )} \log \left (x e + d\right )}{2 \, {\left (b^{5} d^{6} x + a b^{4} d^{6} + {\left (a^{4} b x^{3} + a^{5} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{2} d x^{3} + a^{4} b d x^{2} - a^{5} d x\right )} e^{5} + {\left (6 \, a^{2} b^{3} d^{2} x^{3} - 2 \, a^{3} b^{2} d^{2} x^{2} - 7 \, a^{4} b d^{2} x + a^{5} d^{2}\right )} e^{4} - 4 \, {\left (a b^{4} d^{3} x^{3} - 2 \, a^{2} b^{3} d^{3} x^{2} - 2 \, a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} e^{3} + {\left (b^{5} d^{4} x^{3} - 7 \, a b^{4} d^{4} x^{2} - 2 \, a^{2} b^{3} d^{4} x + 6 \, a^{3} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{5} d^{5} x^{2} - a b^{4} d^{5} x - 2 \, a^{2} b^{3} d^{5}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1066 vs. \(2 (148) = 296\).
time = 2.52, size = 1066, normalized size = 6.79 \begin {gather*} - \frac {b \left (- 3 A b e + 2 B a e + B b d\right ) \log {\left (x + \frac {- 3 A a b^{2} e^{2} - 3 A b^{3} d e + 2 B a^{2} b e^{2} + 3 B a b^{2} d e + B b^{3} d^{2} - \frac {a^{5} b e^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b^{2} d e^{4} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{3} d^{2} e^{3} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{4} d^{3} e^{2} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{5} d^{4} e \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {b^{6} d^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{3} e^{2} + 4 B a b^{2} e^{2} + 2 B b^{3} d e} \right )}}{\left (a e - b d\right )^{4}} + \frac {b \left (- 3 A b e + 2 B a e + B b d\right ) \log {\left (x + \frac {- 3 A a b^{2} e^{2} - 3 A b^{3} d e + 2 B a^{2} b e^{2} + 3 B a b^{2} d e + B b^{3} d^{2} + \frac {a^{5} b e^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b^{2} d e^{4} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{3} d^{2} e^{3} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{4} d^{3} e^{2} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{5} d^{4} e \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {b^{6} d^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{3} e^{2} + 4 B a b^{2} e^{2} + 2 B b^{3} d e} \right )}}{\left (a e - b d\right )^{4}} + \frac {- A a^{2} e^{2} + 5 A a b d e + 2 A b^{2} d^{2} - B a^{2} d e - 5 B a b d^{2} + x^{2} \cdot \left (6 A b^{2} e^{2} - 4 B a b e^{2} - 2 B b^{2} d e\right ) + x \left (3 A a b e^{2} + 9 A b^{2} d e - 2 B a^{2} e^{2} - 7 B a b d e - 3 B b^{2} d^{2}\right )}{2 a^{4} d^{2} e^{3} - 6 a^{3} b d^{3} e^{2} + 6 a^{2} b^{2} d^{4} e - 2 a b^{3} d^{5} + x^{3} \cdot \left (2 a^{3} b e^{5} - 6 a^{2} b^{2} d e^{4} + 6 a b^{3} d^{2} e^{3} - 2 b^{4} d^{3} e^{2}\right ) + x^{2} \cdot \left (2 a^{4} e^{5} - 2 a^{3} b d e^{4} - 6 a^{2} b^{2} d^{2} e^{3} + 10 a b^{3} d^{3} e^{2} - 4 b^{4} d^{4} e\right ) + x \left (4 a^{4} d e^{4} - 10 a^{3} b d^{2} e^{3} + 6 a^{2} b^{2} d^{3} e^{2} + 2 a b^{3} d^{4} e - 2 b^{4} d^{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**2/(e*x+d)**3,x)

[Out]

-b*(-3*A*b*e + 2*B*a*e + B*b*d)*log(x + (-3*A*a*b**2*e**2 - 3*A*b**3*d*e + 2*B*a**2*b*e**2 + 3*B*a*b**2*d*e +
B*b**3*d**2 - a**5*b*e**5*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4 + 5*a**4*b**2*d*e**4*(-3*A*b*e + 2*B*a*e
+ B*b*d)/(a*e - b*d)**4 - 10*a**3*b**3*d**2*e**3*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4 + 10*a**2*b**4*d
**3*e**2*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4 - 5*a*b**5*d**4*e*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d
)**4 + b**6*d**5*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4)/(-6*A*b**3*e**2 + 4*B*a*b**2*e**2 + 2*B*b**3*d*e
))/(a*e - b*d)**4 + b*(-3*A*b*e + 2*B*a*e + B*b*d)*log(x + (-3*A*a*b**2*e**2 - 3*A*b**3*d*e + 2*B*a**2*b*e**2
+ 3*B*a*b**2*d*e + B*b**3*d**2 + a**5*b*e**5*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4 - 5*a**4*b**2*d*e**4*
(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4 + 10*a**3*b**3*d**2*e**3*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)*
*4 - 10*a**2*b**4*d**3*e**2*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4 + 5*a*b**5*d**4*e*(-3*A*b*e + 2*B*a*e
+ B*b*d)/(a*e - b*d)**4 - b**6*d**5*(-3*A*b*e + 2*B*a*e + B*b*d)/(a*e - b*d)**4)/(-6*A*b**3*e**2 + 4*B*a*b**2*
e**2 + 2*B*b**3*d*e))/(a*e - b*d)**4 + (-A*a**2*e**2 + 5*A*a*b*d*e + 2*A*b**2*d**2 - B*a**2*d*e - 5*B*a*b*d**2
+ x**2*(6*A*b**2*e**2 - 4*B*a*b*e**2 - 2*B*b**2*d*e) + x*(3*A*a*b*e**2 + 9*A*b**2*d*e - 2*B*a**2*e**2 - 7*B*a
*b*d*e - 3*B*b**2*d**2))/(2*a**4*d**2*e**3 - 6*a**3*b*d**3*e**2 + 6*a**2*b**2*d**4*e - 2*a*b**3*d**5 + x**3*(2
*a**3*b*e**5 - 6*a**2*b**2*d*e**4 + 6*a*b**3*d**2*e**3 - 2*b**4*d**3*e**2) + x**2*(2*a**4*e**5 - 2*a**3*b*d*e*
*4 - 6*a**2*b**2*d**2*e**3 + 10*a*b**3*d**3*e**2 - 4*b**4*d**4*e) + x*(4*a**4*d*e**4 - 10*a**3*b*d**2*e**3 + 6
*a**2*b**2*d**3*e**2 + 2*a*b**3*d**4*e - 2*b**4*d**5))

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Giac [A]
time = 1.34, size = 304, normalized size = 1.94 \begin {gather*} -\frac {{\left (B b^{3} d + 2 \, B a b^{2} e - 3 \, A b^{3} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac {\frac {B a b^{4}}{b x + a} - \frac {A b^{5}}{b x + a}}{b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}} - \frac {3 \, B b^{2} d e^{2} + 2 \, B a b e^{3} - 5 \, A b^{2} e^{3} + \frac {2 \, {\left (2 \, B b^{4} d^{2} e - B a b^{3} d e^{2} - 3 \, A b^{4} d e^{2} - B a^{2} b^{2} e^{3} + 3 \, A a b^{3} e^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b d - a e\right )}^{4} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.44, size = 454, normalized size = 2.89 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\left (b^2\,\left (3\,A\,e-B\,d\right )-2\,B\,a\,b\,e\right )\,\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4\,\left (B\,b^2\,d-3\,A\,b^2\,e+2\,B\,a\,b\,e\right )}\right )\,\left (b^2\,\left (3\,A\,e-B\,d\right )-2\,B\,a\,b\,e\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {B\,a^2\,d\,e+A\,a^2\,e^2+5\,B\,a\,b\,d^2-5\,A\,a\,b\,d\,e-2\,A\,b^2\,d^2}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {x\,\left (a\,e+3\,b\,d\right )\,\left (2\,B\,a\,e-3\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b\,e\,x^2\,\left (2\,B\,a\,e-3\,A\,b\,e+B\,b\,d\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{x\,\left (b\,d^2+2\,a\,e\,d\right )+a\,d^2+x^2\,\left (a\,e^2+2\,b\,d\,e\right )+b\,e^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)^2*(d + e*x)^3),x)

[Out]

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

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