3.15.41 \(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=120 \[ -\frac {b (d+e x)^6 (-2 a B e-A b e+3 b B d)}{6 e^4}+\frac {(d+e x)^5 (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {(d+e x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4} \]

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Rubi [A]  time = 0.16, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \begin {gather*} -\frac {b (d+e x)^6 (-2 a B e-A b e+3 b B d)}{6 e^4}+\frac {(d+e x)^5 (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {(d+e x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

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

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Mathematica [A]  time = 0.07, size = 216, normalized size = 1.80 \begin {gather*} \frac {1}{4} x^4 \left (a^2 e^2 (A e+3 B d)+6 a b d e (A e+B d)+b^2 d^2 (3 A e+B d)\right )+\frac {1}{3} d x^3 \left (A \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a B d (3 a e+2 b d)\right )+\frac {1}{5} e x^5 \left (a^2 B e^2+2 a b e (A e+3 B d)+3 b^2 d (A e+B d)\right )+a^2 A d^3 x+\frac {1}{2} a d^2 x^2 (3 a A e+a B d+2 A b d)+\frac {1}{6} b e^2 x^6 (2 a B e+A b e+3 b B d)+\frac {1}{7} b^2 B e^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.37, size = 287, normalized size = 2.39 \begin {gather*} \frac {1}{7} x^{7} e^{3} b^{2} B + \frac {1}{2} x^{6} e^{2} d b^{2} B + \frac {1}{3} x^{6} e^{3} b a B + \frac {1}{6} x^{6} e^{3} b^{2} A + \frac {3}{5} x^{5} e d^{2} b^{2} B + \frac {6}{5} x^{5} e^{2} d b a B + \frac {1}{5} x^{5} e^{3} a^{2} B + \frac {3}{5} x^{5} e^{2} d b^{2} A + \frac {2}{5} x^{5} e^{3} b a A + \frac {1}{4} x^{4} d^{3} b^{2} B + \frac {3}{2} x^{4} e d^{2} b a B + \frac {3}{4} x^{4} e^{2} d a^{2} B + \frac {3}{4} x^{4} e d^{2} b^{2} A + \frac {3}{2} x^{4} e^{2} d b a A + \frac {1}{4} x^{4} e^{3} a^{2} A + \frac {2}{3} x^{3} d^{3} b a B + x^{3} e d^{2} a^{2} B + \frac {1}{3} x^{3} d^{3} b^{2} A + 2 x^{3} e d^{2} b a A + x^{3} e^{2} d a^{2} A + \frac {1}{2} x^{2} d^{3} a^{2} B + x^{2} d^{3} b a A + \frac {3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^3*b^2*B + 1/2*x^6*e^2*d*b^2*B + 1/3*x^6*e^3*b*a*B + 1/6*x^6*e^3*b^2*A + 3/5*x^5*e*d^2*b^2*B + 6/5*x^
5*e^2*d*b*a*B + 1/5*x^5*e^3*a^2*B + 3/5*x^5*e^2*d*b^2*A + 2/5*x^5*e^3*b*a*A + 1/4*x^4*d^3*b^2*B + 3/2*x^4*e*d^
2*b*a*B + 3/4*x^4*e^2*d*a^2*B + 3/4*x^4*e*d^2*b^2*A + 3/2*x^4*e^2*d*b*a*A + 1/4*x^4*e^3*a^2*A + 2/3*x^3*d^3*b*
a*B + x^3*e*d^2*a^2*B + 1/3*x^3*d^3*b^2*A + 2*x^3*e*d^2*b*a*A + x^3*e^2*d*a^2*A + 1/2*x^2*d^3*a^2*B + x^2*d^3*
b*a*A + 3/2*x^2*e*d^2*a^2*A + x*d^3*a^2*A

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giac [B]  time = 0.17, size = 281, normalized size = 2.34 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} e^{3} + \frac {1}{2} \, B b^{2} d x^{6} e^{2} + \frac {3}{5} \, B b^{2} d^{2} x^{5} e + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {1}{3} \, B a b x^{6} e^{3} + \frac {1}{6} \, A b^{2} x^{6} e^{3} + \frac {6}{5} \, B a b d x^{5} e^{2} + \frac {3}{5} \, A b^{2} d x^{5} e^{2} + \frac {3}{2} \, B a b d^{2} x^{4} e + \frac {3}{4} \, A b^{2} d^{2} x^{4} e + \frac {2}{3} \, B a b d^{3} x^{3} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {1}{5} \, B a^{2} x^{5} e^{3} + \frac {2}{5} \, A a b x^{5} e^{3} + \frac {3}{4} \, B a^{2} d x^{4} e^{2} + \frac {3}{2} \, A a b d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + 2 \, A a b d^{2} x^{3} e + \frac {1}{2} \, B a^{2} d^{3} x^{2} + A a b d^{3} x^{2} + \frac {1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac {3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b^2*x^7*e^3 + 1/2*B*b^2*d*x^6*e^2 + 3/5*B*b^2*d^2*x^5*e + 1/4*B*b^2*d^3*x^4 + 1/3*B*a*b*x^6*e^3 + 1/6*A*
b^2*x^6*e^3 + 6/5*B*a*b*d*x^5*e^2 + 3/5*A*b^2*d*x^5*e^2 + 3/2*B*a*b*d^2*x^4*e + 3/4*A*b^2*d^2*x^4*e + 2/3*B*a*
b*d^3*x^3 + 1/3*A*b^2*d^3*x^3 + 1/5*B*a^2*x^5*e^3 + 2/5*A*a*b*x^5*e^3 + 3/4*B*a^2*d*x^4*e^2 + 3/2*A*a*b*d*x^4*
e^2 + B*a^2*d^2*x^3*e + 2*A*a*b*d^2*x^3*e + 1/2*B*a^2*d^3*x^2 + A*a*b*d^3*x^2 + 1/4*A*a^2*x^4*e^3 + A*a^2*d*x^
3*e^2 + 3/2*A*a^2*d^2*x^2*e + A*a^2*d^3*x

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maple [B]  time = 0.05, size = 244, normalized size = 2.03 \begin {gather*} \frac {B \,b^{2} e^{3} x^{7}}{7}+A \,a^{2} d^{3} x +\frac {\left (2 B a b \,e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{2}\right ) x^{6}}{6}+\frac {\left (B \,a^{2} e^{3}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) a b +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a b +\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (A \,b^{2} d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{2}+2 \left (3 A \,d^{2} e +B \,d^{3}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.49, size = 236, normalized size = 1.97 \begin {gather*} \frac {1}{7} \, B b^{2} e^{3} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B b^{2} d^{2} e + 3 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{3} + A a^{2} e^{3} + 3 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{2} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 231, normalized size = 1.92 \begin {gather*} x^4\,\left (\frac {3\,B\,a^2\,d\,e^2}{4}+\frac {A\,a^2\,e^3}{4}+\frac {3\,B\,a\,b\,d^2\,e}{2}+\frac {3\,A\,a\,b\,d\,e^2}{2}+\frac {B\,b^2\,d^3}{4}+\frac {3\,A\,b^2\,d^2\,e}{4}\right )+x^3\,\left (B\,a^2\,d^2\,e+A\,a^2\,d\,e^2+\frac {2\,B\,a\,b\,d^3}{3}+2\,A\,a\,b\,d^2\,e+\frac {A\,b^2\,d^3}{3}\right )+x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,b\,d\,e^2}{5}+\frac {2\,A\,a\,b\,e^3}{5}+\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}\right )+A\,a^2\,d^3\,x+\frac {a\,d^2\,x^2\,\left (3\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^2\,x^6\,\left (A\,b\,e+2\,B\,a\,e+3\,B\,b\,d\right )}{6}+\frac {B\,b^2\,e^3\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.11, size = 296, normalized size = 2.47 \begin {gather*} A a^{2} d^{3} x + \frac {B b^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac {A b^{2} e^{3}}{6} + \frac {B a b e^{3}}{3} + \frac {B b^{2} d e^{2}}{2}\right ) + x^{5} \left (\frac {2 A a b e^{3}}{5} + \frac {3 A b^{2} d e^{2}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a b d e^{2}}{5} + \frac {3 B b^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a b d e^{2}}{2} + \frac {3 A b^{2} d^{2} e}{4} + \frac {3 B a^{2} d e^{2}}{4} + \frac {3 B a b d^{2} e}{2} + \frac {B b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + 2 A a b d^{2} e + \frac {A b^{2} d^{3}}{3} + B a^{2} d^{2} e + \frac {2 B a b d^{3}}{3}\right ) + x^{2} \left (\frac {3 A a^{2} d^{2} e}{2} + A a b d^{3} + \frac {B a^{2} d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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