3.12.34 \(\int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx\) [1134]

Optimal. Leaf size=191 \[ \frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) x}{b^5}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^6 (a+b x)}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^2}{2 b^6}+\frac {B e^4 (a+b x)^3}{3 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \log (a+b x)}{b^6} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 191, 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 {e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac {2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac {2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac {B e^4 (a+b x)^3}{3 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(a + b*x)^3,x]

[Out]

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

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Mathematica [A]
time = 0.07, size = 187, normalized size = 0.98 \begin {gather*} \frac {6 b e^2 \left (6 a^2 B e^2-3 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x+3 b^2 e^3 (4 b B d+A b e-3 a B e) x^2+2 b^3 B e^4 x^3-\frac {3 (A b-a B) (b d-a e)^4}{(a+b x)^2}-\frac {6 (b d-a e)^3 (b B d+4 A b e-5 a B e)}{a+b x}+12 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \log (a+b x)}{6 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x)^3,x]

[Out]

(6*b*e^2*(6*a^2*B*e^2 - 3*a*b*e*(4*B*d + A*e) + 2*b^2*d*(3*B*d + 2*A*e))*x + 3*b^2*e^3*(4*b*B*d + A*b*e - 3*a*
B*e)*x^2 + 2*b^3*B*e^4*x^3 - (3*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x)^2 - (6*(b*d - a*e)^3*(b*B*d + 4*A*b*e - 5
*a*B*e))/(a + b*x) + 12*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*Log[a + b*x])/(6*b^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(185)=370\).
time = 0.08, size = 427, normalized size = 2.24

method	result	size
norman	\(\frac {\frac {\left (12 A \,a^{3} b \,e^{4}-24 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -20 B \,a^{4} e^{4}+48 B \,a^{3} b d \,e^{3}-36 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x}{b^{5}}+\frac {18 A \,a^{4} b \,e^{4}-36 A \,a^{3} b^{2} d \,e^{3}+18 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e -A \,b^{5} d^{4}-30 B \,a^{5} e^{4}+72 B \,a^{4} b d \,e^{3}-54 B \,a^{3} b^{2} d^{2} e^{2}+12 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{2 b^{6}}+\frac {B \,e^{4} x^{5}}{3 b}-\frac {2 e^{2} \left (3 A a b \,e^{2}-6 A \,b^{2} d e -5 B \,a^{2} e^{2}+12 B a b d e -9 b^{2} B \,d^{2}\right ) x^{3}}{3 b^{3}}+\frac {e^{3} \left (3 A b e -5 B a e +12 B b d \right ) x^{4}}{6 b^{2}}}{\left (b x +a \right )^{2}}+\frac {2 e \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -5 B \,a^{3} e^{3}+12 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(416\)
default	\(-\frac {e^{2} \left (-\frac {1}{3} b^{2} B \,x^{3} e^{2}-\frac {1}{2} A \,b^{2} e^{2} x^{2}+\frac {3}{2} B a b \,e^{2} x^{2}-2 B \,b^{2} d e \,x^{2}+3 A a b \,e^{2} x -4 A \,b^{2} d e x -6 B \,a^{2} e^{2} x +12 B a b d e x -6 B \,b^{2} d^{2} x \right )}{b^{5}}-\frac {-4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-12 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +5 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-8 B a \,b^{3} d^{3} e +B \,b^{4} d^{4}}{b^{6} \left (b x +a \right )}+\frac {2 e \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -5 B \,a^{3} e^{3}+12 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) \ln \left (b x +a \right )}{b^{6}}-\frac {A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{2 b^{6} \left (b x +a \right )^{2}}\)	\(427\)
risch	\(\frac {e^{4} B \,x^{3}}{3 b^{3}}+\frac {e^{4} A \,x^{2}}{2 b^{3}}-\frac {3 e^{4} B a \,x^{2}}{2 b^{4}}+\frac {2 e^{3} B d \,x^{2}}{b^{3}}-\frac {3 e^{4} A a x}{b^{4}}+\frac {4 e^{3} A d x}{b^{3}}+\frac {6 e^{4} B \,a^{2} x}{b^{5}}-\frac {12 e^{3} B a d x}{b^{4}}+\frac {6 e^{2} B \,d^{2} x}{b^{3}}+\frac {\left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -5 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}-18 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x +\frac {7 A \,a^{4} b \,e^{4}-20 A \,a^{3} b^{2} d \,e^{3}+18 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e -A \,b^{5} d^{4}-9 B \,a^{5} e^{4}+28 B \,a^{4} b d \,e^{3}-30 B \,a^{3} b^{2} d^{2} e^{2}+12 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{2 b}}{b^{5} \left (b x +a \right )^{2}}+\frac {6 e^{4} \ln \left (b x +a \right ) A \,a^{2}}{b^{5}}-\frac {12 e^{3} \ln \left (b x +a \right ) A a d}{b^{4}}+\frac {6 e^{2} \ln \left (b x +a \right ) A \,d^{2}}{b^{3}}-\frac {10 e^{4} \ln \left (b x +a \right ) B \,a^{3}}{b^{6}}+\frac {24 e^{3} \ln \left (b x +a \right ) B \,a^{2} d}{b^{5}}-\frac {18 e^{2} \ln \left (b x +a \right ) B a \,d^{2}}{b^{4}}+\frac {4 e \ln \left (b x +a \right ) B \,d^{3}}{b^{3}}\)	\(475\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-e^2/b^5*(-1/3*b^2*B*x^3*e^2-1/2*A*b^2*e^2*x^2+3/2*B*a*b*e^2*x^2-2*B*b^2*d*e*x^2+3*A*a*b*e^2*x-4*A*b^2*d*e*x-6
*B*a^2*e^2*x+12*B*a*b*d*e*x-6*B*b^2*d^2*x)-(-4*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3-12*A*a*b^3*d^2*e^2+4*A*b^4*d^3*e
+5*B*a^4*e^4-16*B*a^3*b*d*e^3+18*B*a^2*b^2*d^2*e^2-8*B*a*b^3*d^3*e+B*b^4*d^4)/b^6/(b*x+a)+2/b^6*e*(3*A*a^2*b*e
^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e-5*B*a^3*e^3+12*B*a^2*b*d*e^2-9*B*a*b^2*d^2*e+2*B*b^3*d^3)*ln(b*x+a)-1/2*(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/(b*x+a)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (195) = 390\).
time = 0.31, size = 426, normalized size = 2.23 \begin {gather*} -\frac {9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + {\left (B a b^{4} + A b^{5}\right )} d^{4} - 4 \, {\left (3 \, B a^{2} b^{3} e - A a b^{4} e\right )} d^{3} + 6 \, {\left (5 \, B a^{3} b^{2} e^{2} - 3 \, A a^{2} b^{3} e^{2}\right )} d^{2} - 4 \, {\left (7 \, B a^{4} b e^{3} - 5 \, A a^{3} b^{2} e^{3}\right )} d + 2 \, {\left (B b^{5} d^{4} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4} - 4 \, {\left (2 \, B a b^{4} e - A b^{5} e\right )} d^{3} + 6 \, {\left (3 \, B a^{2} b^{3} e^{2} - 2 \, A a b^{4} e^{2}\right )} d^{2} - 4 \, {\left (4 \, B a^{3} b^{2} e^{3} - 3 \, A a^{2} b^{3} e^{3}\right )} d\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, B b^{2} x^{3} e^{4} + 3 \, {\left (4 \, B b^{2} d e^{3} - 3 \, B a b e^{4} + A b^{2} e^{4}\right )} x^{2} + 6 \, {\left (6 \, B b^{2} d^{2} e^{2} + 6 \, B a^{2} e^{4} - 3 \, A a b e^{4} - 4 \, {\left (3 \, B a b e^{3} - A b^{2} e^{3}\right )} d\right )} x}{6 \, b^{5}} + \frac {2 \, {\left (2 \, B b^{3} d^{3} e - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4} - 3 \, {\left (3 \, B a b^{2} e^{2} - A b^{3} e^{2}\right )} d^{2} + 6 \, {\left (2 \, B a^{2} b e^{3} - A a 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)^3,x, algorithm="maxima")

[Out]

-1/2*(9*B*a^5*e^4 - 7*A*a^4*b*e^4 + (B*a*b^4 + A*b^5)*d^4 - 4*(3*B*a^2*b^3*e - A*a*b^4*e)*d^3 + 6*(5*B*a^3*b^2
*e^2 - 3*A*a^2*b^3*e^2)*d^2 - 4*(7*B*a^4*b*e^3 - 5*A*a^3*b^2*e^3)*d + 2*(B*b^5*d^4 + 5*B*a^4*b*e^4 - 4*A*a^3*b
^2*e^4 - 4*(2*B*a*b^4*e - A*b^5*e)*d^3 + 6*(3*B*a^2*b^3*e^2 - 2*A*a*b^4*e^2)*d^2 - 4*(4*B*a^3*b^2*e^3 - 3*A*a^
2*b^3*e^3)*d)*x)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) + 1/6*(2*B*b^2*x^3*e^4 + 3*(4*B*b^2*d*e^3 - 3*B*a*b*e^4 + A*b
^2*e^4)*x^2 + 6*(6*B*b^2*d^2*e^2 + 6*B*a^2*e^4 - 3*A*a*b*e^4 - 4*(3*B*a*b*e^3 - A*b^2*e^3)*d)*x)/b^5 + 2*(2*B*
b^3*d^3*e - 5*B*a^3*e^4 + 3*A*a^2*b*e^4 - 3*(3*B*a*b^2*e^2 - A*b^3*e^2)*d^2 + 6*(2*B*a^2*b*e^3 - A*a*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. 632 vs. \(2 (195) = 390\).
time = 0.76, size = 632, normalized size = 3.31 \begin {gather*} -\frac {6 \, B b^{5} d^{4} x + 3 \, {\left (B a b^{4} + A b^{5}\right )} d^{4} - {\left (2 \, B b^{5} x^{5} - 27 \, B a^{5} + 21 \, A a^{4} b - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \, {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 3 \, {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 6 \, {\left (B a^{4} b + A a^{3} b^{2}\right )} x\right )} e^{4} - 12 \, {\left (B b^{5} d x^{4} - 2 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d x^{3} - {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d x^{2} + 2 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d x + {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d\right )} e^{3} - 18 \, {\left (2 \, B b^{5} d^{2} x^{3} + 4 \, B a b^{4} d^{2} x^{2} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} x - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2}\right )} e^{2} - 12 \, {\left (2 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} x + {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3}\right )} e + 12 \, {\left ({\left (5 \, B a^{5} - 3 \, A a^{4} b + {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} e^{4} - 6 \, {\left ({\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d x^{2} + 2 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d x + {\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d\right )} e^{3} + 3 \, {\left ({\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} x^{2} + 2 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} x + {\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2}\right )} e^{2} - 2 \, {\left (B b^{5} d^{3} x^{2} + 2 \, B a b^{4} d^{3} x + B a^{2} b^{3} d^{3}\right )} e\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/6*(6*B*b^5*d^4*x + 3*(B*a*b^4 + A*b^5)*d^4 - (2*B*b^5*x^5 - 27*B*a^5 + 21*A*a^4*b - (5*B*a*b^4 - 3*A*b^5)*x
^4 + 4*(5*B*a^2*b^3 - 3*A*a*b^4)*x^3 + 3*(21*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 + 6*(B*a^4*b + A*a^3*b^2)*x)*e^4 -
12*(B*b^5*d*x^4 - 2*(2*B*a*b^4 - A*b^5)*d*x^3 - (11*B*a^2*b^3 - 4*A*a*b^4)*d*x^2 + 2*(B*a^3*b^2 - 2*A*a^2*b^3)
*d*x + (7*B*a^4*b - 5*A*a^3*b^2)*d)*e^3 - 18*(2*B*b^5*d^2*x^3 + 4*B*a*b^4*d^2*x^2 - 4*(B*a^2*b^3 - A*a*b^4)*d^
2*x - (5*B*a^3*b^2 - 3*A*a^2*b^3)*d^2)*e^2 - 12*(2*(2*B*a*b^4 - A*b^5)*d^3*x + (3*B*a^2*b^3 - A*a*b^4)*d^3)*e
+ 12*((5*B*a^5 - 3*A*a^4*b + (5*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 2*(5*B*a^4*b - 3*A*a^3*b^2)*x)*e^4 - 6*((2*B*a^
2*b^3 - A*a*b^4)*d*x^2 + 2*(2*B*a^3*b^2 - A*a^2*b^3)*d*x + (2*B*a^4*b - A*a^3*b^2)*d)*e^3 + 3*((3*B*a*b^4 - A*
b^5)*d^2*x^2 + 2*(3*B*a^2*b^3 - A*a*b^4)*d^2*x + (3*B*a^3*b^2 - A*a^2*b^3)*d^2)*e^2 - 2*(B*b^5*d^3*x^2 + 2*B*a
*b^4*d^3*x + B*a^2*b^3*d^3)*e)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (194) = 388\).
time = 5.28, size = 444, normalized size = 2.32 \begin {gather*} \frac {B e^{4} x^{3}}{3 b^{3}} + x^{2} \left (\frac {A e^{4}}{2 b^{3}} - \frac {3 B a e^{4}}{2 b^{4}} + \frac {2 B d e^{3}}{b^{3}}\right ) + x \left (- \frac {3 A a e^{4}}{b^{4}} + \frac {4 A d e^{3}}{b^{3}} + \frac {6 B a^{2} e^{4}}{b^{5}} - \frac {12 B a d e^{3}}{b^{4}} + \frac {6 B d^{2} e^{2}}{b^{3}}\right ) + \frac {7 A a^{4} b e^{4} - 20 A a^{3} b^{2} d e^{3} + 18 A a^{2} b^{3} d^{2} e^{2} - 4 A a b^{4} d^{3} e - A b^{5} d^{4} - 9 B a^{5} e^{4} + 28 B a^{4} b d e^{3} - 30 B a^{3} b^{2} d^{2} e^{2} + 12 B a^{2} b^{3} d^{3} e - B a b^{4} d^{4} + x \left (8 A a^{3} b^{2} e^{4} - 24 A a^{2} b^{3} d e^{3} + 24 A a b^{4} d^{2} e^{2} - 8 A b^{5} d^{3} e - 10 B a^{4} b e^{4} + 32 B a^{3} b^{2} d e^{3} - 36 B a^{2} b^{3} d^{2} e^{2} + 16 B a b^{4} d^{3} e - 2 B b^{5} d^{4}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac {2 e \left (a e - b d\right )^{2} \left (- 3 A b e + 5 B a e - 2 B b d\right ) \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)**3,x)

[Out]

B*e**4*x**3/(3*b**3) + x**2*(A*e**4/(2*b**3) - 3*B*a*e**4/(2*b**4) + 2*B*d*e**3/b**3) + x*(-3*A*a*e**4/b**4 +
4*A*d*e**3/b**3 + 6*B*a**2*e**4/b**5 - 12*B*a*d*e**3/b**4 + 6*B*d**2*e**2/b**3) + (7*A*a**4*b*e**4 - 20*A*a**3
*b**2*d*e**3 + 18*A*a**2*b**3*d**2*e**2 - 4*A*a*b**4*d**3*e - A*b**5*d**4 - 9*B*a**5*e**4 + 28*B*a**4*b*d*e**3
- 30*B*a**3*b**2*d**2*e**2 + 12*B*a**2*b**3*d**3*e - B*a*b**4*d**4 + x*(8*A*a**3*b**2*e**4 - 24*A*a**2*b**3*d
*e**3 + 24*A*a*b**4*d**2*e**2 - 8*A*b**5*d**3*e - 10*B*a**4*b*e**4 + 32*B*a**3*b**2*d*e**3 - 36*B*a**2*b**3*d*
*2*e**2 + 16*B*a*b**4*d**3*e - 2*B*b**5*d**4))/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**2) - 2*e*(a*e - b*d)**2*(
-3*A*b*e + 5*B*a*e - 2*B*b*d)*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. 420 vs. \(2 (195) = 390\).
time = 1.18, size = 420, normalized size = 2.20 \begin {gather*} \frac {2 \, {\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \, {\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, B b^{6} x^{3} e^{4} + 12 \, B b^{6} d x^{2} e^{3} + 36 \, B b^{6} d^{2} x e^{2} - 9 \, B a b^{5} x^{2} e^{4} + 3 \, A b^{6} x^{2} e^{4} - 72 \, B a b^{5} d x e^{3} + 24 \, A b^{6} d x e^{3} + 36 \, B a^{2} b^{4} x e^{4} - 18 \, A a b^{5} x e^{4}}{6 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(b*x+a)^3,x, algorithm="giac")

[Out]

2*(2*B*b^3*d^3*e - 9*B*a*b^2*d^2*e^2 + 3*A*b^3*d^2*e^2 + 12*B*a^2*b*d*e^3 - 6*A*a*b^2*d*e^3 - 5*B*a^3*e^4 + 3*
A*a^2*b*e^4)*log(abs(b*x + a))/b^6 - 1/2*(B*a*b^4*d^4 + A*b^5*d^4 - 12*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 30*
B*a^3*b^2*d^2*e^2 - 18*A*a^2*b^3*d^2*e^2 - 28*B*a^4*b*d*e^3 + 20*A*a^3*b^2*d*e^3 + 9*B*a^5*e^4 - 7*A*a^4*b*e^4
+ 2*(B*b^5*d^4 - 8*B*a*b^4*d^3*e + 4*A*b^5*d^3*e + 18*B*a^2*b^3*d^2*e^2 - 12*A*a*b^4*d^2*e^2 - 16*B*a^3*b^2*d
*e^3 + 12*A*a^2*b^3*d*e^3 + 5*B*a^4*b*e^4 - 4*A*a^3*b^2*e^4)*x)/((b*x + a)^2*b^6) + 1/6*(2*B*b^6*x^3*e^4 + 12*
B*b^6*d*x^2*e^3 + 36*B*b^6*d^2*x*e^2 - 9*B*a*b^5*x^2*e^4 + 3*A*b^6*x^2*e^4 - 72*B*a*b^5*d*x*e^3 + 24*A*b^6*d*x
*e^3 + 36*B*a^2*b^4*x*e^4 - 18*A*a*b^5*x*e^4)/b^9

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Mupad [B]
time = 0.14, size = 451, normalized size = 2.36 \begin {gather*} x^2\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{2\,b^3}-\frac {3\,B\,a\,e^4}{2\,b^4}\right )-\frac {\frac {9\,B\,a^5\,e^4-28\,B\,a^4\,b\,d\,e^3-7\,A\,a^4\,b\,e^4+30\,B\,a^3\,b^2\,d^2\,e^2+20\,A\,a^3\,b^2\,d\,e^3-12\,B\,a^2\,b^3\,d^3\,e-18\,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}{2\,b}+x\,\left (5\,B\,a^4\,e^4-16\,B\,a^3\,b\,d\,e^3-4\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3-8\,B\,a\,b^3\,d^3\,e-12\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}-x\,\left (\frac {3\,a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^3}-\frac {3\,B\,a\,e^4}{b^4}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^3}+\frac {3\,B\,a^2\,e^4}{b^5}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-10\,B\,a^3\,e^4+24\,B\,a^2\,b\,d\,e^3+6\,A\,a^2\,b\,e^4-18\,B\,a\,b^2\,d^2\,e^2-12\,A\,a\,b^2\,d\,e^3+4\,B\,b^3\,d^3\,e+6\,A\,b^3\,d^2\,e^2\right )}{b^6}+\frac {B\,e^4\,x^3}{3\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^4)/(a + b*x)^3,x)

[Out]

x^2*((A*e^4 + 4*B*d*e^3)/(2*b^3) - (3*B*a*e^4)/(2*b^4)) - ((A*b^5*d^4 + 9*B*a^5*e^4 - 7*A*a^4*b*e^4 + B*a*b^4*
d^4 + 20*A*a^3*b^2*d*e^3 - 12*B*a^2*b^3*d^3*e - 18*A*a^2*b^3*d^2*e^2 + 30*B*a^3*b^2*d^2*e^2 + 4*A*a*b^4*d^3*e
- 28*B*a^4*b*d*e^3)/(2*b) + x*(5*B*a^4*e^4 + B*b^4*d^4 - 4*A*a^3*b*e^4 + 4*A*b^4*d^3*e - 12*A*a*b^3*d^2*e^2 +
12*A*a^2*b^2*d*e^3 + 18*B*a^2*b^2*d^2*e^2 - 8*B*a*b^3*d^3*e - 16*B*a^3*b*d*e^3))/(a^2*b^5 + b^7*x^2 + 2*a*b^6*
x) - x*((3*a*((A*e^4 + 4*B*d*e^3)/b^3 - (3*B*a*e^4)/b^4))/b - (2*d*e^2*(2*A*e + 3*B*d))/b^3 + (3*B*a^2*e^4)/b^
5) + (log(a + b*x)*(6*A*a^2*b*e^4 - 10*B*a^3*e^4 + 4*B*b^3*d^3*e + 6*A*b^3*d^2*e^2 - 18*B*a*b^2*d^2*e^2 - 12*A
*a*b^2*d*e^3 + 24*B*a^2*b*d*e^3))/b^6 + (B*e^4*x^3)/(3*b^3)

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