3.11.47 \(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx\) [1047]

Optimal. Leaf size=133 \[ -\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \begin {gather*} \frac {b (a+b x)^4 (-3 a B e+A b e+2 b B d)}{60 e (d+e x)^4 (b d-a e)^3}+\frac {(a+b x)^4 (-3 a B e+A b e+2 b B d)}{15 e (d+e x)^5 (b d-a e)^2}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^7,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

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] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx &=-\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) \int \frac {(a+b x)^3}{(d+e x)^6} \, dx}{3 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {(b (2 b B d+A b e-3 a B e)) \int \frac {(a+b x)^3}{(d+e x)^5} \, dx}{15 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 211, normalized size = 1.59 \begin {gather*} -\frac {2 a^3 e^3 (5 A e+B (d+6 e x))+3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{60 e^5 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^7,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(127)=254\).
time = 0.07, size = 281, normalized size = 2.11

method	result	size
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {b^{2} \left (A b e +3 B a e +2 B b d \right ) x^{3}}{3 e^{2}}-\frac {b \left (3 A a b \,e^{2}+A \,b^{2} d e +3 B \,a^{2} e^{2}+3 B a b d e +2 b^{2} B \,d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (6 A \,a^{2} b \,e^{3}+3 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +2 B \,a^{3} e^{3}+3 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\)	\(266\)
default	\(-\frac {b^{3} B}{2 e^{5} \left (e x +d \right )^{2}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{4 e^{5} \left (e x +d \right )^{4}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{3 e^{5} \left (e x +d \right )^{3}}\)	\(281\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {\left (A \,b^{3} e^{2}+3 B a \,b^{2} e^{2}+2 b^{3} B d e \right ) x^{3}}{3 e^{3}}-\frac {\left (3 A a \,b^{2} e^{3}+A \,b^{3} d \,e^{2}+3 B \,a^{2} b \,e^{3}+3 B a \,b^{2} d \,e^{2}+2 b^{3} B e \,d^{2}\right ) x^{2}}{4 e^{4}}-\frac {\left (6 A \,a^{2} b \,e^{4}+3 A a \,b^{2} d \,e^{3}+A \,b^{3} d^{2} e^{2}+2 B \,a^{3} e^{4}+3 B \,a^{2} b d \,e^{3}+3 B a \,b^{2} d^{2} e^{2}+2 b^{3} B e \,d^{3}\right ) x}{10 e^{5}}-\frac {10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(294\)
gosper	\(-\frac {30 b^{3} B \,x^{4} e^{4}+20 A \,b^{3} e^{4} x^{3}+60 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}+15 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}+45 B a \,b^{2} d \,e^{3} x^{2}+30 B \,b^{3} d^{2} e^{2} x^{2}+36 A \,a^{2} b \,e^{4} x +18 A a \,b^{2} d \,e^{3} x +6 A \,b^{3} d^{2} e^{2} x +12 B \,a^{3} e^{4} x +18 B \,a^{2} b d \,e^{3} x +18 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{3} e x +10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\)	\(300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (135) = 270\).
time = 0.30, size = 313, normalized size = 2.35 \begin {gather*} -\frac {30 \, B b^{3} x^{4} e^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4}\right )} x^{3} + 3 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + 3 \, B a^{2} b e^{4} + 3 \, A a b^{2} e^{4} + {\left (3 \, B a b^{2} e^{3} + A b^{3} e^{3}\right )} d\right )} x^{2} + 2 \, {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d + 6 \, {\left (2 \, B b^{3} d^{3} e + 2 \, B a^{3} e^{4} + 6 \, A a^{2} b e^{4} + {\left (3 \, B a b^{2} e^{2} + A b^{3} e^{2}\right )} d^{2} + 3 \, {\left (B a^{2} b e^{3} + A a b^{2} e^{3}\right )} d\right )} x}{60 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/60*(30*B*b^3*x^4*e^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2*e + A*b^3*e)*d^3 + 20*(2*B*b^3*d*e^3 + 3*B*a
*b^2*e^4 + A*b^3*e^4)*x^3 + 3*(B*a^2*b*e^2 + A*a*b^2*e^2)*d^2 + 15*(2*B*b^3*d^2*e^2 + 3*B*a^2*b*e^4 + 3*A*a*b^
2*e^4 + (3*B*a*b^2*e^3 + A*b^3*e^3)*d)*x^2 + 2*(B*a^3*e^3 + 3*A*a^2*b*e^3)*d + 6*(2*B*b^3*d^3*e + 2*B*a^3*e^4
+ 6*A*a^2*b*e^4 + (3*B*a*b^2*e^2 + A*b^3*e^2)*d^2 + 3*(B*a^2*b*e^3 + A*a*b^2*e^3)*d)*x)/(x^6*e^11 + 6*d*x^5*e^
10 + 15*d^2*x^4*e^9 + 20*d^3*x^3*e^8 + 15*d^4*x^2*e^7 + 6*d^5*x*e^6 + d^6*e^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (135) = 270\).
time = 0.57, size = 294, normalized size = 2.21 \begin {gather*} -\frac {2 \, B b^{3} d^{4} + {\left (30 \, B b^{3} x^{4} + 10 \, A a^{3} + 20 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 12 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} + {\left (40 \, B b^{3} d x^{3} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d x + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 3 \, {\left (10 \, B b^{3} d^{2} x^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} + {\left (12 \, B b^{3} d^{3} x + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e}{60 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/60*(2*B*b^3*d^4 + (30*B*b^3*x^4 + 10*A*a^3 + 20*(3*B*a*b^2 + A*b^3)*x^3 + 45*(B*a^2*b + A*a*b^2)*x^2 + 12*(
B*a^3 + 3*A*a^2*b)*x)*e^4 + (40*B*b^3*d*x^3 + 15*(3*B*a*b^2 + A*b^3)*d*x^2 + 18*(B*a^2*b + A*a*b^2)*d*x + 2*(B
*a^3 + 3*A*a^2*b)*d)*e^3 + 3*(10*B*b^3*d^2*x^2 + 2*(3*B*a*b^2 + A*b^3)*d^2*x + (B*a^2*b + A*a*b^2)*d^2)*e^2 +
(12*B*b^3*d^3*x + (3*B*a*b^2 + A*b^3)*d^3)*e)/(x^6*e^11 + 6*d*x^5*e^10 + 15*d^2*x^4*e^9 + 20*d^3*x^3*e^8 + 15*
d^4*x^2*e^7 + 6*d^5*x*e^6 + d^6*e^5)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**7,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (135) = 270\).
time = 1.17, size = 282, normalized size = 2.12 \begin {gather*} -\frac {{\left (30 \, B b^{3} x^{4} e^{4} + 40 \, B b^{3} d x^{3} e^{3} + 30 \, B b^{3} d^{2} x^{2} e^{2} + 12 \, B b^{3} d^{3} x e + 2 \, B b^{3} d^{4} + 60 \, B a b^{2} x^{3} e^{4} + 20 \, A b^{3} x^{3} e^{4} + 45 \, B a b^{2} d x^{2} e^{3} + 15 \, A b^{3} d x^{2} e^{3} + 18 \, B a b^{2} d^{2} x e^{2} + 6 \, A b^{3} d^{2} x e^{2} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 45 \, B a^{2} b x^{2} e^{4} + 45 \, A a b^{2} x^{2} e^{4} + 18 \, B a^{2} b d x e^{3} + 18 \, A a b^{2} d x e^{3} + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} + 12 \, B a^{3} x e^{4} + 36 \, A a^{2} b x e^{4} + 2 \, B a^{3} d e^{3} + 6 \, A a^{2} b d e^{3} + 10 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(30*B*b^3*x^4*e^4 + 40*B*b^3*d*x^3*e^3 + 30*B*b^3*d^2*x^2*e^2 + 12*B*b^3*d^3*x*e + 2*B*b^3*d^4 + 60*B*a*
b^2*x^3*e^4 + 20*A*b^3*x^3*e^4 + 45*B*a*b^2*d*x^2*e^3 + 15*A*b^3*d*x^2*e^3 + 18*B*a*b^2*d^2*x*e^2 + 6*A*b^3*d^
2*x*e^2 + 3*B*a*b^2*d^3*e + A*b^3*d^3*e + 45*B*a^2*b*x^2*e^4 + 45*A*a*b^2*x^2*e^4 + 18*B*a^2*b*d*x*e^3 + 18*A*
a*b^2*d*x*e^3 + 3*B*a^2*b*d^2*e^2 + 3*A*a*b^2*d^2*e^2 + 12*B*a^3*x*e^4 + 36*A*a^2*b*x*e^4 + 2*B*a^3*d*e^3 + 6*
A*a^2*b*d*e^3 + 10*A*a^3*e^4)*e^(-5)/(x*e + d)^6

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Mupad [B]
time = 1.15, size = 321, normalized size = 2.41 \begin {gather*} -\frac {\frac {2\,B\,a^3\,d\,e^3+10\,A\,a^3\,e^4+3\,B\,a^2\,b\,d^2\,e^2+6\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+3\,A\,a\,b^2\,d^2\,e^2+2\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{60\,e^5}+\frac {x\,\left (2\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2+6\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+3\,A\,a\,b^2\,d\,e^2+2\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{10\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{3\,e^2}+\frac {b\,x^2\,\left (3\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+3\,A\,a\,b\,e^2+2\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^3\,x^4}{2\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^7,x)

[Out]

-((10*A*a^3*e^4 + 2*B*b^3*d^4 + A*b^3*d^3*e + 2*B*a^3*d*e^3 + 3*A*a*b^2*d^2*e^2 + 3*B*a^2*b*d^2*e^2 + 6*A*a^2*
b*d*e^3 + 3*B*a*b^2*d^3*e)/(60*e^5) + (x*(2*B*a^3*e^3 + 2*B*b^3*d^3 + 6*A*a^2*b*e^3 + A*b^3*d^2*e + 3*A*a*b^2*
d*e^2 + 3*B*a*b^2*d^2*e + 3*B*a^2*b*d*e^2))/(10*e^4) + (b^2*x^3*(A*b*e + 3*B*a*e + 2*B*b*d))/(3*e^2) + (b*x^2*
(3*B*a^2*e^2 + 2*B*b^2*d^2 + 3*A*a*b*e^2 + A*b^2*d*e + 3*B*a*b*d*e))/(4*e^3) + (B*b^3*x^4)/(2*e))/(d^6 + e^6*x
^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)

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