3.10.7 \(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=86 \[ \frac {(a+b x)^4 (-5 a B e+A b e+4 b B d)}{20 e (d+e x)^4 (b d-a e)^2}-\frac {(a+b x)^4 (B d-A e)}{5 e (d+e x)^5 (b d-a e)} \]

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Rubi [A]  time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {(a+b x)^4 (-5 a B e+A b e+4 b B d)}{20 e (d+e x)^4 (b d-a e)^2}-\frac {(a+b x)^4 (B d-A e)}{5 e (d+e x)^5 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^6,x]

[Out]

-((B*d - A*e)*(a + b*x)^4)/(5*e*(b*d - a*e)*(d + e*x)^5) + ((4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^4)/(20*e*(b*
d - a*e)^2*(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 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)^3 (A+B x)}{(d+e x)^6} \, dx &=-\frac {(B d-A e) (a+b x)^4}{5 e (b d-a e) (d+e x)^5}+\frac {(4 b B d+A b e-5 a B e) \int \frac {(a+b x)^3}{(d+e x)^5} \, dx}{5 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^4}{5 e (b d-a e) (d+e x)^5}+\frac {(4 b B d+A b e-5 a B e) (a+b x)^4}{20 e (b d-a e)^2 (d+e x)^4}\\ \end {align*}

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Mathematica [B]  time = 0.10, size = 211, normalized size = 2.45 \begin {gather*} -\frac {a^3 e^3 (4 A e+B (d+5 e x))+a^2 b e^2 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+a b^2 e \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )}{20 e^5 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^6,x]

[Out]

-1/20*(a^3*e^3*(4*A*e + B*(d + 5*e*x)) + a^2*b*e^2*(3*A*e*(d + 5*e*x) + 2*B*(d^2 + 5*d*e*x + 10*e^2*x^2)) + a*
b^2*e*(2*A*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) + b^3*(A*e*(d^3
+ 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*B*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))
)/(e^5*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^3*(A + B*x))/(d + e*x)^6,x]

[Out]

IntegrateAlgebraic[((a + b*x)^3*(A + B*x))/(d + e*x)^6, x]

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fricas [B]  time = 0.76, size = 304, normalized size = 3.53 \begin {gather*} -\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.20, size = 281, normalized size = 3.27 \begin {gather*} -\frac {{\left (20 \, B b^{3} x^{4} e^{4} + 40 \, B b^{3} d x^{3} e^{3} + 40 \, B b^{3} d^{2} x^{2} e^{2} + 20 \, B b^{3} d^{3} x e + 4 \, B b^{3} d^{4} + 30 \, B a b^{2} x^{3} e^{4} + 10 \, A b^{3} x^{3} e^{4} + 30 \, B a b^{2} d x^{2} e^{3} + 10 \, A b^{3} d x^{2} e^{3} + 15 \, B a b^{2} d^{2} x e^{2} + 5 \, A b^{3} d^{2} x e^{2} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 20 \, B a^{2} b x^{2} e^{4} + 20 \, A a b^{2} x^{2} e^{4} + 10 \, B a^{2} b d x e^{3} + 10 \, A a b^{2} d x e^{3} + 2 \, B a^{2} b d^{2} e^{2} + 2 \, A a b^{2} d^{2} e^{2} + 5 \, B a^{3} x e^{4} + 15 \, A a^{2} b x e^{4} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} + 4 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{20 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^6,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.01, size = 281, normalized size = 3.27 \begin {gather*} -\frac {B \,b^{3}}{\left (e x +d \right ) e^{5}}-\frac {\left (A b e +3 B a e -4 B b d \right ) b^{2}}{2 \left (e x +d \right )^{2} e^{5}}-\frac {\left (A a b \,e^{2}-A d \,b^{2} e +B \,a^{2} e^{2}-3 B d a b e +2 B \,b^{2} d^{2}\right ) b}{\left (e x +d \right )^{3} e^{5}}-\frac {3 A \,a^{2} b \,e^{3}-6 A d a \,b^{2} e^{2}+3 A \,d^{2} b^{3} e +B \,a^{3} e^{3}-6 B d \,a^{2} b \,e^{2}+9 B \,d^{2} a \,b^{2} e -4 B \,b^{3} d^{3}}{4 \left (e x +d \right )^{4} e^{5}}-\frac {A \,a^{3} e^{4}-3 A d \,a^{2} b \,e^{3}+3 A \,d^{2} a \,b^{2} e^{2}-A \,d^{3} b^{3} e -B d \,a^{3} e^{3}+3 B \,d^{2} a^{2} b \,e^{2}-3 B \,d^{3} a \,b^{2} e +B \,b^{3} d^{4}}{5 \left (e x +d \right )^{5} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^6,x)

[Out]

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

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maxima [B]  time = 0.57, size = 304, normalized size = 3.53 \begin {gather*} -\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} 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)^6,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.12, size = 307, normalized size = 3.57 \begin {gather*} -\frac {\frac {B\,a^3\,d\,e^3+4\,A\,a^3\,e^4+2\,B\,a^2\,b\,d^2\,e^2+3\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+2\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{20\,e^5}+\frac {x\,\left (B\,a^3\,e^3+2\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+4\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{2\,e^3}+\frac {B\,b^3\,x^4}{e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^6,x)

[Out]

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

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sympy [B]  time = 68.78, size = 372, normalized size = 4.33 \begin {gather*} \frac {- 4 A a^{3} e^{4} - 3 A a^{2} b d e^{3} - 2 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} - 2 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e - 4 B b^{3} d^{4} - 20 B b^{3} e^{4} x^{4} + x^{3} \left (- 10 A b^{3} e^{4} - 30 B a b^{2} e^{4} - 40 B b^{3} d e^{3}\right ) + x^{2} \left (- 20 A a b^{2} e^{4} - 10 A b^{3} d e^{3} - 20 B a^{2} b e^{4} - 30 B a b^{2} d e^{3} - 40 B b^{3} d^{2} e^{2}\right ) + x \left (- 15 A a^{2} b e^{4} - 10 A a b^{2} d e^{3} - 5 A b^{3} d^{2} e^{2} - 5 B a^{3} e^{4} - 10 B a^{2} b d e^{3} - 15 B a b^{2} d^{2} e^{2} - 20 B b^{3} d^{3} e\right )}{20 d^{5} e^{5} + 100 d^{4} e^{6} x + 200 d^{3} e^{7} x^{2} + 200 d^{2} e^{8} x^{3} + 100 d e^{9} x^{4} + 20 e^{10} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**6,x)

[Out]

(-4*A*a**3*e**4 - 3*A*a**2*b*d*e**3 - 2*A*a*b**2*d**2*e**2 - A*b**3*d**3*e - B*a**3*d*e**3 - 2*B*a**2*b*d**2*e
**2 - 3*B*a*b**2*d**3*e - 4*B*b**3*d**4 - 20*B*b**3*e**4*x**4 + x**3*(-10*A*b**3*e**4 - 30*B*a*b**2*e**4 - 40*
B*b**3*d*e**3) + x**2*(-20*A*a*b**2*e**4 - 10*A*b**3*d*e**3 - 20*B*a**2*b*e**4 - 30*B*a*b**2*d*e**3 - 40*B*b**
3*d**2*e**2) + x*(-15*A*a**2*b*e**4 - 10*A*a*b**2*d*e**3 - 5*A*b**3*d**2*e**2 - 5*B*a**3*e**4 - 10*B*a**2*b*d*
e**3 - 15*B*a*b**2*d**2*e**2 - 20*B*b**3*d**3*e))/(20*d**5*e**5 + 100*d**4*e**6*x + 200*d**3*e**7*x**2 + 200*d
**2*e**8*x**3 + 100*d*e**9*x**4 + 20*e**10*x**5)

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