3.1683 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=193 \[ -\frac {b^3 (d+e x)^2 (-4 a B e-A b e+5 b B d)}{2 e^6}+\frac {2 b^2 x (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^5}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 (d+e x)}+\frac {(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac {2 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)}{e^6}+\frac {b^4 B (d+e x)^3}{3 e^6} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac {b^3 (d+e x)^2 (-4 a B e-A b e+5 b B d)}{2 e^6}+\frac {2 b^2 x (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^5}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 (d+e x)}+\frac {(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac {2 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)}{e^6}+\frac {b^4 B (d+e x)^3}{3 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.10, size = 187, normalized size = 0.97 \[ \frac {-6 b^2 e x \left (-6 a^2 B e^2-4 a b e (A e-3 B d)+3 b^2 d (A e-2 B d)\right )+3 b^3 e^2 x^2 (4 a B e+A b e-3 b B d)-\frac {6 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{d+e x}+\frac {3 (b d-a e)^4 (B d-A e)}{(d+e x)^2}-12 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)+2 b^4 B e^3 x^3}{6 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.87, size = 652, normalized size = 3.38 \[ \frac {2 \, B b^{4} e^{5} x^{5} - 27 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 21 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 30 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 18 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - {\left (5 \, B b^{4} d e^{4} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B b^{4} d^{2} e^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \, {\left (21 \, B b^{4} d^{3} e^{2} - 11 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 8 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 6 \, {\left (B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 12 \, {\left (5 \, B b^{4} d^{5} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (5 \, B b^{4} d^{3} e^{2} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 2 \, {\left (5 \, B b^{4} d^{4} e - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*B*b^4*e^5*x^5 - 27*B*b^4*d^5 - 3*A*a^4*e^5 + 21*(4*B*a*b^3 + A*b^4)*d^4*e - 30*(3*B*a^2*b^2 + 2*A*a*b^3
)*d^3*e^2 + 18*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - 3*(B*a^4 + 4*A*a^3*b)*d*e^4 - (5*B*b^4*d*e^4 - 3*(4*B*a*b^3
+ A*b^4)*e^5)*x^4 + 4*(5*B*b^4*d^2*e^3 - 3*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 +
3*(21*B*b^4*d^3*e^2 - 11*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 8*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4)*x^2 + 6*(B*b^4*d^4*
e + (4*B*a*b^3 + A*b^4)*d^3*e^2 - 4*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B
*a^4 + 4*A*a^3*b)*e^5)*x - 12*(5*B*b^4*d^5 - 3*(4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2
- (2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (5*B*b^4*d^3*e^2 - 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A
*a*b^3)*d*e^4 - (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 2*(5*B*b^4*d^4*e - 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B
*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - (2*B*a^3*b + 3*A*a^2*b^2)*d*e^4)*x)*log(e*x + d))/(e^8*x^2 + 2*d*e^7*x + d^2*e
^6)

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giac [B]  time = 0.17, size = 418, normalized size = 2.17 \[ -2 \, {\left (5 \, B b^{4} d^{3} - 12 \, B a b^{3} d^{2} e - 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} + 6 \, A a b^{3} d e^{2} - 2 \, B a^{3} b e^{3} - 3 \, A a^{2} b^{2} e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, B b^{4} x^{3} e^{6} - 9 \, B b^{4} d x^{2} e^{5} + 36 \, B b^{4} d^{2} x e^{4} + 12 \, B a b^{3} x^{2} e^{6} + 3 \, A b^{4} x^{2} e^{6} - 72 \, B a b^{3} d x e^{5} - 18 \, A b^{4} d x e^{5} + 36 \, B a^{2} b^{2} x e^{6} + 24 \, A a b^{3} x e^{6}\right )} e^{\left (-9\right )} - \frac {{\left (9 \, B b^{4} d^{5} - 28 \, B a b^{3} d^{4} e - 7 \, A b^{4} d^{4} e + 30 \, B a^{2} b^{2} d^{3} e^{2} + 20 \, A a b^{3} d^{3} e^{2} - 12 \, B a^{3} b d^{2} e^{3} - 18 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} + A a^{4} e^{5} + 2 \, {\left (5 \, B b^{4} d^{4} e - 16 \, B a b^{3} d^{3} e^{2} - 4 \, A b^{4} d^{3} e^{2} + 18 \, B a^{2} b^{2} d^{2} e^{3} + 12 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} + B a^{4} e^{5} + 4 \, A a^{3} b e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \, {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 601, normalized size = 3.11 \[ \frac {B \,b^{4} x^{3}}{3 e^{3}}-\frac {A \,a^{4}}{2 \left (e x +d \right )^{2} e}+\frac {2 A \,a^{3} b d}{\left (e x +d \right )^{2} e^{2}}-\frac {3 A \,a^{2} b^{2} d^{2}}{\left (e x +d \right )^{2} e^{3}}+\frac {2 A a \,b^{3} d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {A \,b^{4} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {A \,b^{4} x^{2}}{2 e^{3}}+\frac {B \,a^{4} d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {2 B \,a^{3} b \,d^{2}}{\left (e x +d \right )^{2} e^{3}}+\frac {3 B \,a^{2} b^{2} d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {2 B a \,b^{3} d^{4}}{\left (e x +d \right )^{2} e^{5}}+\frac {2 B a \,b^{3} x^{2}}{e^{3}}+\frac {B \,b^{4} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {3 B \,b^{4} d \,x^{2}}{2 e^{4}}-\frac {4 A \,a^{3} b}{\left (e x +d \right ) e^{2}}+\frac {12 A \,a^{2} b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {6 A \,a^{2} b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {12 A a \,b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {12 A a \,b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {4 A a \,b^{3} x}{e^{3}}+\frac {4 A \,b^{4} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 A \,b^{4} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 A \,b^{4} d x}{e^{4}}-\frac {B \,a^{4}}{\left (e x +d \right ) e^{2}}+\frac {8 B \,a^{3} b d}{\left (e x +d \right ) e^{3}}+\frac {4 B \,a^{3} b \ln \left (e x +d \right )}{e^{3}}-\frac {18 B \,a^{2} b^{2} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {18 B \,a^{2} b^{2} d \ln \left (e x +d \right )}{e^{4}}+\frac {6 B \,a^{2} b^{2} x}{e^{3}}+\frac {16 B a \,b^{3} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {24 B a \,b^{3} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {12 B a \,b^{3} d x}{e^{4}}-\frac {5 B \,b^{4} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {10 B \,b^{4} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {6 B \,b^{4} d^{2} x}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^4/e^3*B*x^3+1/2*b^4/e^3*A*x^2-1/e^2/(e*x+d)*B*a^4-1/2/e/(e*x+d)^2*A*a^4-5/e^6/(e*x+d)*B*b^4*d^4-4/e^2/(e
*x+d)*A*a^3*b+6*b^2/e^3*ln(e*x+d)*A*a^2-12*b^3/e^4*B*x*a*d-3/e^3/(e*x+d)^2*A*d^2*a^2*b^2+2/e^4/(e*x+d)^2*A*a*b
^3*d^3-2/e^3/(e*x+d)^2*B*d^2*a^3*b+3/e^4/(e*x+d)^2*B*a^2*b^2*d^3-2/e^5/(e*x+d)^2*B*a*b^3*d^4+24*b^3/e^5*ln(e*x
+d)*B*a*d^2+12/e^3/(e*x+d)*A*a^2*b^2*d-12/e^4/(e*x+d)*A*a*b^3*d^2+8/e^3/(e*x+d)*B*a^3*b*d-18/e^4/(e*x+d)*B*a^2
*b^2*d^2+16/e^5/(e*x+d)*B*a*b^3*d^3+2/e^2/(e*x+d)^2*A*d*a^3*b-12*b^3/e^4*ln(e*x+d)*A*a*d-18*b^2/e^4*ln(e*x+d)*
B*a^2*d+1/2/e^6/(e*x+d)^2*B*b^4*d^5+4/e^5/(e*x+d)*A*b^4*d^3+6*b^4/e^5*ln(e*x+d)*A*d^2+4*b/e^3*ln(e*x+d)*B*a^3-
10*b^4/e^6*ln(e*x+d)*B*d^3+4*b^3/e^3*A*x*a-3*b^4/e^4*A*x*d+6*b^2/e^3*B*x*a^2+6*b^4/e^5*B*x*d^2+2*b^3/e^3*B*x^2
*a-3/2*b^4/e^4*B*x^2*d-1/2/e^5/(e*x+d)^2*A*b^4*d^4+1/2/e^2/(e*x+d)^2*B*d*a^4

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maxima [B]  time = 0.69, size = 419, normalized size = 2.17 \[ -\frac {9 \, B b^{4} d^{5} + A a^{4} e^{5} - 7 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 6 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 2 \, {\left (5 \, B b^{4} d^{4} e - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {2 \, B b^{4} e^{2} x^{3} - 3 \, {\left (3 \, B b^{4} d e - {\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{2} + 6 \, {\left (6 \, B b^{4} d^{2} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac {2 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.05, size = 451, normalized size = 2.34 \[ x^2\,\left (\frac {A\,b^4+4\,B\,a\,b^3}{2\,e^3}-\frac {3\,B\,b^4\,d}{2\,e^4}\right )-\frac {\frac {B\,a^4\,d\,e^4+A\,a^4\,e^5-12\,B\,a^3\,b\,d^2\,e^3+4\,A\,a^3\,b\,d\,e^4+30\,B\,a^2\,b^2\,d^3\,e^2-18\,A\,a^2\,b^2\,d^2\,e^3-28\,B\,a\,b^3\,d^4\,e+20\,A\,a\,b^3\,d^3\,e^2+9\,B\,b^4\,d^5-7\,A\,b^4\,d^4\,e}{2\,e}+x\,\left (B\,a^4\,e^4-8\,B\,a^3\,b\,d\,e^3+4\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2-12\,A\,a^2\,b^2\,d\,e^3-16\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4-4\,A\,b^4\,d^3\,e\right )}{d^2\,e^5+2\,d\,e^6\,x+e^7\,x^2}-x\,\left (\frac {3\,d\,\left (\frac {A\,b^4+4\,B\,a\,b^3}{e^3}-\frac {3\,B\,b^4\,d}{e^4}\right )}{e}-\frac {2\,a\,b^2\,\left (2\,A\,b+3\,B\,a\right )}{e^3}+\frac {3\,B\,b^4\,d^2}{e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (4\,B\,a^3\,b\,e^3-18\,B\,a^2\,b^2\,d\,e^2+6\,A\,a^2\,b^2\,e^3+24\,B\,a\,b^3\,d^2\,e-12\,A\,a\,b^3\,d\,e^2-10\,B\,b^4\,d^3+6\,A\,b^4\,d^2\,e\right )}{e^6}+\frac {B\,b^4\,x^3}{3\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 7.27, size = 444, normalized size = 2.30 \[ \frac {B b^{4} x^{3}}{3 e^{3}} + \frac {2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right ) \log {\left (d + e x \right )}}{e^{6}} + x^{2} \left (\frac {A b^{4}}{2 e^{3}} + \frac {2 B a b^{3}}{e^{3}} - \frac {3 B b^{4} d}{2 e^{4}}\right ) + x \left (\frac {4 A a b^{3}}{e^{3}} - \frac {3 A b^{4} d}{e^{4}} + \frac {6 B a^{2} b^{2}}{e^{3}} - \frac {12 B a b^{3} d}{e^{4}} + \frac {6 B b^{4} d^{2}}{e^{5}}\right ) + \frac {- A a^{4} e^{5} - 4 A a^{3} b d e^{4} + 18 A a^{2} b^{2} d^{2} e^{3} - 20 A a b^{3} d^{3} e^{2} + 7 A b^{4} d^{4} e - B a^{4} d e^{4} + 12 B a^{3} b d^{2} e^{3} - 30 B a^{2} b^{2} d^{3} e^{2} + 28 B a b^{3} d^{4} e - 9 B b^{4} d^{5} + x \left (- 8 A a^{3} b e^{5} + 24 A a^{2} b^{2} d e^{4} - 24 A a b^{3} d^{2} e^{3} + 8 A b^{4} d^{3} e^{2} - 2 B a^{4} e^{5} + 16 B a^{3} b d e^{4} - 36 B a^{2} b^{2} d^{2} e^{3} + 32 B a b^{3} d^{3} e^{2} - 10 B b^{4} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**4*x**3/(3*e**3) + 2*b*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e - 5*B*b*d)*log(d + e*x)/e**6 + x**2*(A*b**4/(2*e*
*3) + 2*B*a*b**3/e**3 - 3*B*b**4*d/(2*e**4)) + x*(4*A*a*b**3/e**3 - 3*A*b**4*d/e**4 + 6*B*a**2*b**2/e**3 - 12*
B*a*b**3*d/e**4 + 6*B*b**4*d**2/e**5) + (-A*a**4*e**5 - 4*A*a**3*b*d*e**4 + 18*A*a**2*b**2*d**2*e**3 - 20*A*a*
b**3*d**3*e**2 + 7*A*b**4*d**4*e - B*a**4*d*e**4 + 12*B*a**3*b*d**2*e**3 - 30*B*a**2*b**2*d**3*e**2 + 28*B*a*b
**3*d**4*e - 9*B*b**4*d**5 + x*(-8*A*a**3*b*e**5 + 24*A*a**2*b**2*d*e**4 - 24*A*a*b**3*d**2*e**3 + 8*A*b**4*d*
*3*e**2 - 2*B*a**4*e**5 + 16*B*a**3*b*d*e**4 - 36*B*a**2*b**2*d**2*e**3 + 32*B*a*b**3*d**3*e**2 - 10*B*b**4*d*
*4*e))/(2*d**2*e**6 + 4*d*e**7*x + 2*e**8*x**2)

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