3.12.12 \(\int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx\) [1112]

Optimal. Leaf size=155 \[ \frac {(A b-a B) e (b d-a e)^3 x}{b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^2}{2 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^3}{3 b^3}+\frac {(A b-a B) (d+e x)^4}{4 b^2}+\frac {B (d+e x)^5}{5 b e}+\frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 155, 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)^4 \log (a+b x)}{b^6}+\frac {e x (A b-a B) (b d-a e)^3}{b^5}+\frac {(d+e x)^2 (A b-a B) (b d-a e)^2}{2 b^4}+\frac {(d+e x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {(d+e x)^4 (A b-a B)}{4 b^2}+\frac {B (d+e x)^5}{5 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*e*(b*d - a*e)^3*x)/b^5 + ((A*b - a*B)*(b*d - a*e)^2*(d + e*x)^2)/(2*b^4) + ((A*b - a*B)*(b*d - a*
e)*(d + e*x)^3)/(3*b^3) + ((A*b - a*B)*(d + e*x)^4)/(4*b^2) + (B*(d + e*x)^5)/(5*b*e) + ((A*b - a*B)*(b*d - a*
e)^4*Log[a + b*x])/b^6

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)^4}{a+b x} \, dx &=\int \left (\frac {(A b-a B) e (b d-a e)^3}{b^5}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)}+\frac {(A b-a B) e (b d-a e)^2 (d+e x)}{b^4}+\frac {(A b-a B) e (b d-a e) (d+e x)^2}{b^3}+\frac {(A b-a B) e (d+e x)^3}{b^2}+\frac {B (d+e x)^4}{b}\right ) \, dx\\ &=\frac {(A b-a B) e (b d-a e)^3 x}{b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^2}{2 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^3}{3 b^3}+\frac {(A b-a B) (d+e x)^4}{4 b^2}+\frac {B (d+e x)^5}{5 b e}+\frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 257, normalized size = 1.66 \begin {gather*} \frac {b x \left (60 a^4 B e^4-30 a^3 b e^3 (8 B d+2 A e+B e x)+10 a^2 b^2 e^2 \left (3 A e (8 d+e x)+2 B \left (18 d^2+6 d e x+e^2 x^2\right )\right )-5 a b^3 e \left (4 A e \left (18 d^2+6 d e x+e^2 x^2\right )+B \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+b^4 \left (5 A e \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+12 B \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )\right )\right )+60 (A b-a B) (b d-a e)^4 \log (a+b x)}{60 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(60*a^4*B*e^4 - 30*a^3*b*e^3*(8*B*d + 2*A*e + B*e*x) + 10*a^2*b^2*e^2*(3*A*e*(8*d + e*x) + 2*B*(18*d^2 +
6*d*e*x + e^2*x^2)) - 5*a*b^3*e*(4*A*e*(18*d^2 + 6*d*e*x + e^2*x^2) + B*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 +
3*e^3*x^3)) + b^4*(5*A*e*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) + 12*B*(5*d^4 + 10*d^3*e*x + 10*d^2*
e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4))) + 60*(A*b - a*B)*(b*d - a*e)^4*Log[a + b*x])/(60*b^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(147)=294\).
time = 0.08, size = 464, normalized size = 2.99

method	result	size
norman	\(-\frac {\left (A \,a^{3} b \,e^{4}-4 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -B \,a^{4} e^{4}+4 B \,a^{3} b d \,e^{3}-6 B \,a^{2} b^{2} d^{2} e^{2}+4 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x}{b^{5}}+\frac {B \,e^{4} x^{5}}{5 b}+\frac {e \left (A \,a^{2} b \,e^{3}-4 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e -B \,a^{3} e^{3}+4 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x^{2}}{2 b^{4}}-\frac {e^{2} \left (A a b \,e^{2}-4 A \,b^{2} d e -B \,a^{2} e^{2}+4 B a b d e -6 b^{2} B \,d^{2}\right ) x^{3}}{3 b^{3}}+\frac {e^{3} \left (A b e -B a e +4 B b d \right ) x^{4}}{4 b^{2}}+\frac {\left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(404\)
default	\(-\frac {-4 A \,a^{2} b^{2} d \,e^{3} x +6 A a \,b^{3} d^{2} e^{2} x +4 B \,a^{3} b d \,e^{3} x -6 B \,a^{2} b^{2} d^{2} e^{2} x +4 B a \,b^{3} d^{3} e x -2 B \,a^{2} b^{2} d \,e^{3} x^{2}+3 B a \,b^{3} d^{2} e^{2} x^{2}+2 A a \,b^{3} d \,e^{3} x^{2}+\frac {4}{3} B a \,b^{3} d \,e^{3} x^{3}-\frac {1}{5} b^{4} B \,x^{5} e^{4}-\frac {1}{4} A \,b^{4} e^{4} x^{4}-B \,a^{4} e^{4} x -B \,b^{4} d^{4} x -3 A \,b^{4} d^{2} e^{2} x^{2}+\frac {1}{2} B \,a^{3} b \,e^{4} x^{2}+\frac {1}{4} B a \,b^{3} e^{4} x^{4}-B \,b^{4} d \,e^{3} x^{4}+\frac {1}{3} A a \,b^{3} e^{4} x^{3}-\frac {4}{3} A \,b^{4} d \,e^{3} x^{3}-\frac {1}{3} B \,a^{2} b^{2} e^{4} x^{3}-2 B \,b^{4} d^{2} e^{2} x^{3}-\frac {1}{2} A \,a^{2} b^{2} e^{4} x^{2}-2 B \,b^{4} d^{3} e \,x^{2}+A \,a^{3} b \,e^{4} x -4 A \,b^{4} d^{3} e x}{b^{5}}+\frac {\left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(464\)
risch	\(\frac {4 A \,a^{2} d \,e^{3} x}{b^{3}}-\frac {6 A a \,d^{2} e^{2} x}{b^{2}}-\frac {4 B \,a^{3} d \,e^{3} x}{b^{4}}+\frac {6 B \,a^{2} d^{2} e^{2} x}{b^{3}}-\frac {4 B a \,d^{3} e x}{b^{2}}+\frac {2 B \,a^{2} d \,e^{3} x^{2}}{b^{3}}-\frac {3 B a \,d^{2} e^{2} x^{2}}{b^{2}}-\frac {2 A a d \,e^{3} x^{2}}{b^{2}}+\frac {A \,e^{4} x^{4}}{4 b}+\frac {B \,d^{4} x}{b}+\frac {\ln \left (b x +a \right ) A \,d^{4}}{b}+\frac {B \,e^{4} x^{5}}{5 b}-\frac {B \,a^{3} e^{4} x^{2}}{2 b^{4}}-\frac {B a \,e^{4} x^{4}}{4 b^{2}}+\frac {B d \,e^{3} x^{4}}{b}-\frac {A a \,e^{4} x^{3}}{3 b^{2}}+\frac {4 A d \,e^{3} x^{3}}{3 b}+\frac {B \,a^{2} e^{4} x^{3}}{3 b^{3}}+\frac {2 B \,d^{2} e^{2} x^{3}}{b}+\frac {A \,a^{2} e^{4} x^{2}}{2 b^{3}}+\frac {2 B \,d^{3} e \,x^{2}}{b}-\frac {A \,a^{3} e^{4} x}{b^{4}}+\frac {4 A \,d^{3} e x}{b}+\frac {\ln \left (b x +a \right ) A \,a^{4} e^{4}}{b^{5}}-\frac {\ln \left (b x +a \right ) B \,a^{5} e^{4}}{b^{6}}-\frac {\ln \left (b x +a \right ) B a \,d^{4}}{b^{2}}-\frac {4 B a d \,e^{3} x^{3}}{3 b^{2}}-\frac {4 \ln \left (b x +a \right ) A \,a^{3} d \,e^{3}}{b^{4}}+\frac {6 \ln \left (b x +a \right ) A \,a^{2} d^{2} e^{2}}{b^{3}}-\frac {4 \ln \left (b x +a \right ) A a \,d^{3} e}{b^{2}}+\frac {4 \ln \left (b x +a \right ) B \,a^{4} d \,e^{3}}{b^{5}}-\frac {6 \ln \left (b x +a \right ) B \,a^{3} d^{2} e^{2}}{b^{4}}+\frac {4 \ln \left (b x +a \right ) B \,a^{2} d^{3} e}{b^{3}}+\frac {B \,a^{4} e^{4} x}{b^{5}}+\frac {3 A \,d^{2} e^{2} x^{2}}{b}\)	\(521\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (157) = 314\).
time = 0.30, size = 404, normalized size = 2.61 \begin {gather*} \frac {12 \, B b^{4} x^{5} e^{4} + 15 \, {\left (4 \, B b^{4} d e^{3} - B a b^{3} e^{4} + A b^{4} e^{4}\right )} x^{4} + 20 \, {\left (6 \, B b^{4} d^{2} e^{2} + B a^{2} b^{2} e^{4} - A a b^{3} e^{4} - 4 \, {\left (B a b^{3} e^{3} - A b^{4} e^{3}\right )} d\right )} x^{3} + 30 \, {\left (4 \, B b^{4} d^{3} e - B a^{3} b e^{4} + A a^{2} b^{2} e^{4} - 6 \, {\left (B a b^{3} e^{2} - A b^{4} e^{2}\right )} d^{2} + 4 \, {\left (B a^{2} b^{2} e^{3} - A a b^{3} e^{3}\right )} d\right )} x^{2} + 60 \, {\left (B b^{4} d^{4} + B a^{4} e^{4} - A a^{3} b e^{4} - 4 \, {\left (B a b^{3} e - A b^{4} e\right )} d^{3} + 6 \, {\left (B a^{2} b^{2} e^{2} - A a b^{3} e^{2}\right )} d^{2} - 4 \, {\left (B a^{3} b e^{3} - A a^{2} b^{2} e^{3}\right )} d\right )} x}{60 \, b^{5}} - \frac {{\left (B a^{5} e^{4} - A a^{4} b e^{4} + {\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} e - A a b^{4} e\right )} d^{3} + 6 \, {\left (B a^{3} b^{2} e^{2} - A a^{2} b^{3} e^{2}\right )} d^{2} - 4 \, {\left (B a^{4} b e^{3} - A a^{3} b^{2} e^{3}\right )} d\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*x^5*e^4 + 15*(4*B*b^4*d*e^3 - B*a*b^3*e^4 + A*b^4*e^4)*x^4 + 20*(6*B*b^4*d^2*e^2 + B*a^2*b^2*e^
4 - A*a*b^3*e^4 - 4*(B*a*b^3*e^3 - A*b^4*e^3)*d)*x^3 + 30*(4*B*b^4*d^3*e - B*a^3*b*e^4 + A*a^2*b^2*e^4 - 6*(B*
a*b^3*e^2 - A*b^4*e^2)*d^2 + 4*(B*a^2*b^2*e^3 - A*a*b^3*e^3)*d)*x^2 + 60*(B*b^4*d^4 + B*a^4*e^4 - A*a^3*b*e^4
- 4*(B*a*b^3*e - A*b^4*e)*d^3 + 6*(B*a^2*b^2*e^2 - A*a*b^3*e^2)*d^2 - 4*(B*a^3*b*e^3 - A*a^2*b^2*e^3)*d)*x)/b^
5 - (B*a^5*e^4 - A*a^4*b*e^4 + (B*a*b^4 - A*b^5)*d^4 - 4*(B*a^2*b^3*e - A*a*b^4*e)*d^3 + 6*(B*a^3*b^2*e^2 - A*
a^2*b^3*e^2)*d^2 - 4*(B*a^4*b*e^3 - A*a^3*b^2*e^3)*d)*log(b*x + a)/b^6

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (157) = 314\).
time = 0.91, size = 394, normalized size = 2.54 \begin {gather*} \frac {60 \, B b^{5} d^{4} x + {\left (12 \, B b^{5} x^{5} - 15 \, {\left (B a b^{4} - A b^{5}\right )} x^{4} + 20 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 30 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 60 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} x\right )} e^{4} + 20 \, {\left (3 \, B b^{5} d x^{4} - 4 \, {\left (B a b^{4} - A b^{5}\right )} d x^{3} + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d x^{2} - 12 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d x\right )} e^{3} + 60 \, {\left (2 \, B b^{5} d^{2} x^{3} - 3 \, {\left (B a b^{4} - A b^{5}\right )} d^{2} x^{2} + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} x\right )} e^{2} + 120 \, {\left (B b^{5} d^{3} x^{2} - 2 \, {\left (B a b^{4} - A b^{5}\right )} d^{3} x\right )} e - 60 \, {\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*B*b^5*d^4*x + (12*B*b^5*x^5 - 15*(B*a*b^4 - A*b^5)*x^4 + 20*(B*a^2*b^3 - A*a*b^4)*x^3 - 30*(B*a^3*b^2
- A*a^2*b^3)*x^2 + 60*(B*a^4*b - A*a^3*b^2)*x)*e^4 + 20*(3*B*b^5*d*x^4 - 4*(B*a*b^4 - A*b^5)*d*x^3 + 6*(B*a^2
*b^3 - A*a*b^4)*d*x^2 - 12*(B*a^3*b^2 - A*a^2*b^3)*d*x)*e^3 + 60*(2*B*b^5*d^2*x^3 - 3*(B*a*b^4 - A*b^5)*d^2*x^
2 + 6*(B*a^2*b^3 - A*a*b^4)*d^2*x)*e^2 + 120*(B*b^5*d^3*x^2 - 2*(B*a*b^4 - A*b^5)*d^3*x)*e - 60*((B*a*b^4 - A*
b^5)*d^4 - 4*(B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 - 4*(B*a^4*b - A*a^3*b^2)*d*e^3 +
(B*a^5 - A*a^4*b)*e^4)*log(b*x + a))/b^6

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (136) = 272\).
time = 0.62, size = 352, normalized size = 2.27 \begin {gather*} \frac {B e^{4} x^{5}}{5 b} + x^{4} \left (\frac {A e^{4}}{4 b} - \frac {B a e^{4}}{4 b^{2}} + \frac {B d e^{3}}{b}\right ) + x^{3} \left (- \frac {A a e^{4}}{3 b^{2}} + \frac {4 A d e^{3}}{3 b} + \frac {B a^{2} e^{4}}{3 b^{3}} - \frac {4 B a d e^{3}}{3 b^{2}} + \frac {2 B d^{2} e^{2}}{b}\right ) + x^{2} \left (\frac {A a^{2} e^{4}}{2 b^{3}} - \frac {2 A a d e^{3}}{b^{2}} + \frac {3 A d^{2} e^{2}}{b} - \frac {B a^{3} e^{4}}{2 b^{4}} + \frac {2 B a^{2} d e^{3}}{b^{3}} - \frac {3 B a d^{2} e^{2}}{b^{2}} + \frac {2 B d^{3} e}{b}\right ) + x \left (- \frac {A a^{3} e^{4}}{b^{4}} + \frac {4 A a^{2} d e^{3}}{b^{3}} - \frac {6 A a d^{2} e^{2}}{b^{2}} + \frac {4 A d^{3} e}{b} + \frac {B a^{4} e^{4}}{b^{5}} - \frac {4 B a^{3} d e^{3}}{b^{4}} + \frac {6 B a^{2} d^{2} e^{2}}{b^{3}} - \frac {4 B a d^{3} e}{b^{2}} + \frac {B d^{4}}{b}\right ) - \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (157) = 314\).
time = 2.69, size = 444, normalized size = 2.86 \begin {gather*} \frac {12 \, B b^{4} x^{5} e^{4} + 60 \, B b^{4} d x^{4} e^{3} + 120 \, B b^{4} d^{2} x^{3} e^{2} + 120 \, B b^{4} d^{3} x^{2} e + 60 \, B b^{4} d^{4} x - 15 \, B a b^{3} x^{4} e^{4} + 15 \, A b^{4} x^{4} e^{4} - 80 \, B a b^{3} d x^{3} e^{3} + 80 \, A b^{4} d x^{3} e^{3} - 180 \, B a b^{3} d^{2} x^{2} e^{2} + 180 \, A b^{4} d^{2} x^{2} e^{2} - 240 \, B a b^{3} d^{3} x e + 240 \, A b^{4} d^{3} x e + 20 \, B a^{2} b^{2} x^{3} e^{4} - 20 \, A a b^{3} x^{3} e^{4} + 120 \, B a^{2} b^{2} d x^{2} e^{3} - 120 \, A a b^{3} d x^{2} e^{3} + 360 \, B a^{2} b^{2} d^{2} x e^{2} - 360 \, A a b^{3} d^{2} x e^{2} - 30 \, B a^{3} b x^{2} e^{4} + 30 \, A a^{2} b^{2} x^{2} e^{4} - 240 \, B a^{3} b d x e^{3} + 240 \, A a^{2} b^{2} d x e^{3} + 60 \, B a^{4} x e^{4} - 60 \, A a^{3} b x e^{4}}{60 \, b^{5}} - \frac {{\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*x^5*e^4 + 60*B*b^4*d*x^4*e^3 + 120*B*b^4*d^2*x^3*e^2 + 120*B*b^4*d^3*x^2*e + 60*B*b^4*d^4*x - 1
5*B*a*b^3*x^4*e^4 + 15*A*b^4*x^4*e^4 - 80*B*a*b^3*d*x^3*e^3 + 80*A*b^4*d*x^3*e^3 - 180*B*a*b^3*d^2*x^2*e^2 + 1
80*A*b^4*d^2*x^2*e^2 - 240*B*a*b^3*d^3*x*e + 240*A*b^4*d^3*x*e + 20*B*a^2*b^2*x^3*e^4 - 20*A*a*b^3*x^3*e^4 + 1
20*B*a^2*b^2*d*x^2*e^3 - 120*A*a*b^3*d*x^2*e^3 + 360*B*a^2*b^2*d^2*x*e^2 - 360*A*a*b^3*d^2*x*e^2 - 30*B*a^3*b*
x^2*e^4 + 30*A*a^2*b^2*x^2*e^4 - 240*B*a^3*b*d*x*e^3 + 240*A*a^2*b^2*d*x*e^3 + 60*B*a^4*x*e^4 - 60*A*a^3*b*x*e
^4)/b^5 - (B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*d^2*e^2 - 6*A*a^2*b^3*d
^2*e^2 - 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4 - A*a^4*b*e^4)*log(abs(b*x + a))/b^6

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Mupad [B]
time = 1.13, size = 411, normalized size = 2.65 \begin {gather*} x\,\left (\frac {B\,d^4+4\,A\,e\,d^3}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b}\right )}{b}+\frac {2\,d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{3\,b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{3\,b}\right )+x^4\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{4\,b}-\frac {B\,a\,e^4}{4\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b}\right )}{2\,b}+\frac {d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4\right )}{b^6}+\frac {B\,e^4\,x^5}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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