3.16.86 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ -\frac {2 b^5 (d+e x)^{9/2} (-6 a B e-A b e+7 b B d)}{9 e^8}+\frac {6 b^4 (d+e x)^{7/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8}-\frac {2 b^3 (d+e x)^{5/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8}+\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8}-\frac {6 b \sqrt {d+e x} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}-\frac {2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^6 B (d+e x)^{11/2}}{11 e^8} \]

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Rubi [A]  time = 0.15, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 77} \begin {gather*} -\frac {2 b^5 (d+e x)^{9/2} (-6 a B e-A b e+7 b B d)}{9 e^8}+\frac {6 b^4 (d+e x)^{7/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8}-\frac {2 b^3 (d+e x)^{5/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8}+\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8}-\frac {6 b \sqrt {d+e x} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}-\frac {2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^6 B (d+e x)^{11/2}}{11 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(b*d - a*e)^6*(B*d - A*e))/(3*e^8*(d + e*x)^(3/2)) - (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(e^8*Sqr
t[d + e*x]) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*Sqrt[d + e*x])/e^8 + (10*b^2*(b*d - a*e)^3*(7*b
*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(3/2))/(3*e^8) - (2*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e
*x)^(5/2))/e^8 + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (2*b^5*(7*b*B*d -
A*b*e - 6*a*B*e)*(d + e*x)^(9/2))/(9*e^8) + (2*b^6*B*(d + e*x)^(11/2))/(11*e^8)

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 )^3}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{5/2}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^{3/2}}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 \sqrt {d+e x}}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) \sqrt {d+e x}}{e^7}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)^{3/2}}{e^7}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)^{5/2}}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{7/2}}{e^7}+\frac {b^6 B (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^{3/2}}-\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{e^8 \sqrt {d+e x}}-\frac {6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \sqrt {d+e x}}{e^8}+\frac {10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{3/2}}{3 e^8}-\frac {2 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{5/2}}{e^8}+\frac {6 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{7/2}}{7 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{9/2}}{9 e^8}+\frac {2 b^6 B (d+e x)^{11/2}}{11 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 259, normalized size = 0.86 \begin {gather*} \frac {2 \left (-77 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+297 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-693 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+1155 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-2079 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)-693 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+231 (b d-a e)^6 (B d-A e)+63 b^6 B (d+e x)^7\right )}{693 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(231*(b*d - a*e)^6*(B*d - A*e) - 693*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x) - 2079*b*(b*d - a*
e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 1155*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x
)^3 - 693*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 297*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e
- 5*a*B*e)*(d + e*x)^5 - 77*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 63*b^6*B*(d + e*x)^7))/(693*e^8*(d
+ e*x)^(3/2))

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IntegrateAlgebraic [B]  time = 0.23, size = 1069, normalized size = 3.54 \begin {gather*} \frac {2 \left (231 b^6 B d^7-231 A b^6 e d^6-1386 a b^5 B e d^6-4851 b^6 B (d+e x) d^6+1386 a A b^5 e^2 d^5+3465 a^2 b^4 B e^2 d^5-14553 b^6 B (d+e x)^2 d^5+4158 A b^6 e (d+e x) d^5+24948 a b^5 B e (d+e x) d^5-3465 a^2 A b^4 e^3 d^4-4620 a^3 b^3 B e^3 d^4+8085 b^6 B (d+e x)^3 d^4+10395 A b^6 e (d+e x)^2 d^4+62370 a b^5 B e (d+e x)^2 d^4-20790 a A b^5 e^2 (d+e x) d^4-51975 a^2 b^4 B e^2 (d+e x) d^4+4620 a^3 A b^3 e^4 d^3+3465 a^4 b^2 B e^4 d^3-4851 b^6 B (d+e x)^4 d^3-4620 A b^6 e (d+e x)^3 d^3-27720 a b^5 B e (d+e x)^3 d^3-41580 a A b^5 e^2 (d+e x)^2 d^3-103950 a^2 b^4 B e^2 (d+e x)^2 d^3+41580 a^2 A b^4 e^3 (d+e x) d^3+55440 a^3 b^3 B e^3 (d+e x) d^3-3465 a^4 A b^2 e^5 d^2-1386 a^5 b B e^5 d^2+2079 b^6 B (d+e x)^5 d^2+2079 A b^6 e (d+e x)^4 d^2+12474 a b^5 B e (d+e x)^4 d^2+13860 a A b^5 e^2 (d+e x)^3 d^2+34650 a^2 b^4 B e^2 (d+e x)^3 d^2+62370 a^2 A b^4 e^3 (d+e x)^2 d^2+83160 a^3 b^3 B e^3 (d+e x)^2 d^2-41580 a^3 A b^3 e^4 (d+e x) d^2-31185 a^4 b^2 B e^4 (d+e x) d^2+1386 a^5 A b e^6 d+231 a^6 B e^6 d-539 b^6 B (d+e x)^6 d-594 A b^6 e (d+e x)^5 d-3564 a b^5 B e (d+e x)^5 d-4158 a A b^5 e^2 (d+e x)^4 d-10395 a^2 b^4 B e^2 (d+e x)^4 d-13860 a^2 A b^4 e^3 (d+e x)^3 d-18480 a^3 b^3 B e^3 (d+e x)^3 d-41580 a^3 A b^3 e^4 (d+e x)^2 d-31185 a^4 b^2 B e^4 (d+e x)^2 d+20790 a^4 A b^2 e^5 (d+e x) d+8316 a^5 b B e^5 (d+e x) d-231 a^6 A e^7+63 b^6 B (d+e x)^7+77 A b^6 e (d+e x)^6+462 a b^5 B e (d+e x)^6+594 a A b^5 e^2 (d+e x)^5+1485 a^2 b^4 B e^2 (d+e x)^5+2079 a^2 A b^4 e^3 (d+e x)^4+2772 a^3 b^3 B e^3 (d+e x)^4+4620 a^3 A b^3 e^4 (d+e x)^3+3465 a^4 b^2 B e^4 (d+e x)^3+10395 a^4 A b^2 e^5 (d+e x)^2+4158 a^5 b B e^5 (d+e x)^2-4158 a^5 A b e^6 (d+e x)-693 a^6 B e^6 (d+e x)\right )}{693 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(231*b^6*B*d^7 - 231*A*b^6*d^6*e - 1386*a*b^5*B*d^6*e + 1386*a*A*b^5*d^5*e^2 + 3465*a^2*b^4*B*d^5*e^2 - 346
5*a^2*A*b^4*d^4*e^3 - 4620*a^3*b^3*B*d^4*e^3 + 4620*a^3*A*b^3*d^3*e^4 + 3465*a^4*b^2*B*d^3*e^4 - 3465*a^4*A*b^
2*d^2*e^5 - 1386*a^5*b*B*d^2*e^5 + 1386*a^5*A*b*d*e^6 + 231*a^6*B*d*e^6 - 231*a^6*A*e^7 - 4851*b^6*B*d^6*(d +
e*x) + 4158*A*b^6*d^5*e*(d + e*x) + 24948*a*b^5*B*d^5*e*(d + e*x) - 20790*a*A*b^5*d^4*e^2*(d + e*x) - 51975*a^
2*b^4*B*d^4*e^2*(d + e*x) + 41580*a^2*A*b^4*d^3*e^3*(d + e*x) + 55440*a^3*b^3*B*d^3*e^3*(d + e*x) - 41580*a^3*
A*b^3*d^2*e^4*(d + e*x) - 31185*a^4*b^2*B*d^2*e^4*(d + e*x) + 20790*a^4*A*b^2*d*e^5*(d + e*x) + 8316*a^5*b*B*d
*e^5*(d + e*x) - 4158*a^5*A*b*e^6*(d + e*x) - 693*a^6*B*e^6*(d + e*x) - 14553*b^6*B*d^5*(d + e*x)^2 + 10395*A*
b^6*d^4*e*(d + e*x)^2 + 62370*a*b^5*B*d^4*e*(d + e*x)^2 - 41580*a*A*b^5*d^3*e^2*(d + e*x)^2 - 103950*a^2*b^4*B
*d^3*e^2*(d + e*x)^2 + 62370*a^2*A*b^4*d^2*e^3*(d + e*x)^2 + 83160*a^3*b^3*B*d^2*e^3*(d + e*x)^2 - 41580*a^3*A
*b^3*d*e^4*(d + e*x)^2 - 31185*a^4*b^2*B*d*e^4*(d + e*x)^2 + 10395*a^4*A*b^2*e^5*(d + e*x)^2 + 4158*a^5*b*B*e^
5*(d + e*x)^2 + 8085*b^6*B*d^4*(d + e*x)^3 - 4620*A*b^6*d^3*e*(d + e*x)^3 - 27720*a*b^5*B*d^3*e*(d + e*x)^3 +
13860*a*A*b^5*d^2*e^2*(d + e*x)^3 + 34650*a^2*b^4*B*d^2*e^2*(d + e*x)^3 - 13860*a^2*A*b^4*d*e^3*(d + e*x)^3 -
18480*a^3*b^3*B*d*e^3*(d + e*x)^3 + 4620*a^3*A*b^3*e^4*(d + e*x)^3 + 3465*a^4*b^2*B*e^4*(d + e*x)^3 - 4851*b^6
*B*d^3*(d + e*x)^4 + 2079*A*b^6*d^2*e*(d + e*x)^4 + 12474*a*b^5*B*d^2*e*(d + e*x)^4 - 4158*a*A*b^5*d*e^2*(d +
e*x)^4 - 10395*a^2*b^4*B*d*e^2*(d + e*x)^4 + 2079*a^2*A*b^4*e^3*(d + e*x)^4 + 2772*a^3*b^3*B*e^3*(d + e*x)^4 +
2079*b^6*B*d^2*(d + e*x)^5 - 594*A*b^6*d*e*(d + e*x)^5 - 3564*a*b^5*B*d*e*(d + e*x)^5 + 594*a*A*b^5*e^2*(d +
e*x)^5 + 1485*a^2*b^4*B*e^2*(d + e*x)^5 - 539*b^6*B*d*(d + e*x)^6 + 77*A*b^6*e*(d + e*x)^6 + 462*a*b^5*B*e*(d
+ e*x)^6 + 63*b^6*B*(d + e*x)^7))/(693*e^8*(d + e*x)^(3/2))

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fricas [B]  time = 0.45, size = 790, normalized size = 2.62 \begin {gather*} \frac {2 \, {\left (63 \, B b^{6} e^{7} x^{7} - 14336 \, B b^{6} d^{7} - 231 \, A a^{6} e^{7} + 11264 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e - 25344 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 29568 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 18480 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 5544 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - 462 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 7 \, {\left (14 \, B b^{6} d e^{6} - 11 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 3 \, {\left (56 \, B b^{6} d^{2} e^{5} - 44 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 99 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 3 \, {\left (112 \, B b^{6} d^{3} e^{4} - 88 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 198 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 231 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + {\left (896 \, B b^{6} d^{4} e^{3} - 704 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 1584 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 1848 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 1155 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - 3 \, {\left (1792 \, B b^{6} d^{5} e^{2} - 1408 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 3168 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 3696 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 2310 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 693 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} - 3 \, {\left (7168 \, B b^{6} d^{6} e - 5632 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 12672 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 14784 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 9240 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 2772 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 231 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \end {gather*}

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)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^6*e^7*x^7 - 14336*B*b^6*d^7 - 231*A*a^6*e^7 + 11264*(6*B*a*b^5 + A*b^6)*d^6*e - 25344*(5*B*a^2*b
^4 + 2*A*a*b^5)*d^5*e^2 + 29568*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 18480*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^
4 + 5544*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 462*(B*a^6 + 6*A*a^5*b)*d*e^6 - 7*(14*B*b^6*d*e^6 - 11*(6*B*a*b^5
+ A*b^6)*e^7)*x^6 + 3*(56*B*b^6*d^2*e^5 - 44*(6*B*a*b^5 + A*b^6)*d*e^6 + 99*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^
5 - 3*(112*B*b^6*d^3*e^4 - 88*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 198*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 231*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + (896*B*b^6*d^4*e^3 - 704*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 1584*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^2*e^5 - 1848*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 1155*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 3*(1792*
B*b^6*d^5*e^2 - 1408*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 3168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 3696*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d^2*e^5 + 2310*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 693*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 - 3*(
7168*B*b^6*d^6*e - 5632*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 12672*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 14784*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 9240*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 2772*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6
+ 231*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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giac [B]  time = 0.31, size = 1103, normalized size = 3.65

result too large to display

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)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/693*(63*(x*e + d)^(11/2)*B*b^6*e^80 - 539*(x*e + d)^(9/2)*B*b^6*d*e^80 + 2079*(x*e + d)^(7/2)*B*b^6*d^2*e^80
- 4851*(x*e + d)^(5/2)*B*b^6*d^3*e^80 + 8085*(x*e + d)^(3/2)*B*b^6*d^4*e^80 - 14553*sqrt(x*e + d)*B*b^6*d^5*e
^80 + 462*(x*e + d)^(9/2)*B*a*b^5*e^81 + 77*(x*e + d)^(9/2)*A*b^6*e^81 - 3564*(x*e + d)^(7/2)*B*a*b^5*d*e^81 -
594*(x*e + d)^(7/2)*A*b^6*d*e^81 + 12474*(x*e + d)^(5/2)*B*a*b^5*d^2*e^81 + 2079*(x*e + d)^(5/2)*A*b^6*d^2*e^
81 - 27720*(x*e + d)^(3/2)*B*a*b^5*d^3*e^81 - 4620*(x*e + d)^(3/2)*A*b^6*d^3*e^81 + 62370*sqrt(x*e + d)*B*a*b^
5*d^4*e^81 + 10395*sqrt(x*e + d)*A*b^6*d^4*e^81 + 1485*(x*e + d)^(7/2)*B*a^2*b^4*e^82 + 594*(x*e + d)^(7/2)*A*
a*b^5*e^82 - 10395*(x*e + d)^(5/2)*B*a^2*b^4*d*e^82 - 4158*(x*e + d)^(5/2)*A*a*b^5*d*e^82 + 34650*(x*e + d)^(3
/2)*B*a^2*b^4*d^2*e^82 + 13860*(x*e + d)^(3/2)*A*a*b^5*d^2*e^82 - 103950*sqrt(x*e + d)*B*a^2*b^4*d^3*e^82 - 41
580*sqrt(x*e + d)*A*a*b^5*d^3*e^82 + 2772*(x*e + d)^(5/2)*B*a^3*b^3*e^83 + 2079*(x*e + d)^(5/2)*A*a^2*b^4*e^83
- 18480*(x*e + d)^(3/2)*B*a^3*b^3*d*e^83 - 13860*(x*e + d)^(3/2)*A*a^2*b^4*d*e^83 + 83160*sqrt(x*e + d)*B*a^3
*b^3*d^2*e^83 + 62370*sqrt(x*e + d)*A*a^2*b^4*d^2*e^83 + 3465*(x*e + d)^(3/2)*B*a^4*b^2*e^84 + 4620*(x*e + d)^
(3/2)*A*a^3*b^3*e^84 - 31185*sqrt(x*e + d)*B*a^4*b^2*d*e^84 - 41580*sqrt(x*e + d)*A*a^3*b^3*d*e^84 + 4158*sqrt
(x*e + d)*B*a^5*b*e^85 + 10395*sqrt(x*e + d)*A*a^4*b^2*e^85)*e^(-88) - 2/3*(21*(x*e + d)*B*b^6*d^6 - B*b^6*d^7
- 108*(x*e + d)*B*a*b^5*d^5*e - 18*(x*e + d)*A*b^6*d^5*e + 6*B*a*b^5*d^6*e + A*b^6*d^6*e + 225*(x*e + d)*B*a^
2*b^4*d^4*e^2 + 90*(x*e + d)*A*a*b^5*d^4*e^2 - 15*B*a^2*b^4*d^5*e^2 - 6*A*a*b^5*d^5*e^2 - 240*(x*e + d)*B*a^3*
b^3*d^3*e^3 - 180*(x*e + d)*A*a^2*b^4*d^3*e^3 + 20*B*a^3*b^3*d^4*e^3 + 15*A*a^2*b^4*d^4*e^3 + 135*(x*e + d)*B*
a^4*b^2*d^2*e^4 + 180*(x*e + d)*A*a^3*b^3*d^2*e^4 - 15*B*a^4*b^2*d^3*e^4 - 20*A*a^3*b^3*d^3*e^4 - 36*(x*e + d)
*B*a^5*b*d*e^5 - 90*(x*e + d)*A*a^4*b^2*d*e^5 + 6*B*a^5*b*d^2*e^5 + 15*A*a^4*b^2*d^2*e^5 + 3*(x*e + d)*B*a^6*e
^6 + 18*(x*e + d)*A*a^5*b*e^6 - B*a^6*d*e^6 - 6*A*a^5*b*d*e^6 + A*a^6*e^7)*e^(-8)/(x*e + d)^(3/2)

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maple [B]  time = 0.06, size = 913, normalized size = 3.02 \begin {gather*} -\frac {2 \left (-63 B \,b^{6} x^{7} e^{7}-77 A \,b^{6} e^{7} x^{6}-462 B a \,b^{5} e^{7} x^{6}+98 B \,b^{6} d \,e^{6} x^{6}-594 A a \,b^{5} e^{7} x^{5}+132 A \,b^{6} d \,e^{6} x^{5}-1485 B \,a^{2} b^{4} e^{7} x^{5}+792 B a \,b^{5} d \,e^{6} x^{5}-168 B \,b^{6} d^{2} e^{5} x^{5}-2079 A \,a^{2} b^{4} e^{7} x^{4}+1188 A a \,b^{5} d \,e^{6} x^{4}-264 A \,b^{6} d^{2} e^{5} x^{4}-2772 B \,a^{3} b^{3} e^{7} x^{4}+2970 B \,a^{2} b^{4} d \,e^{6} x^{4}-1584 B a \,b^{5} d^{2} e^{5} x^{4}+336 B \,b^{6} d^{3} e^{4} x^{4}-4620 A \,a^{3} b^{3} e^{7} x^{3}+5544 A \,a^{2} b^{4} d \,e^{6} x^{3}-3168 A a \,b^{5} d^{2} e^{5} x^{3}+704 A \,b^{6} d^{3} e^{4} x^{3}-3465 B \,a^{4} b^{2} e^{7} x^{3}+7392 B \,a^{3} b^{3} d \,e^{6} x^{3}-7920 B \,a^{2} b^{4} d^{2} e^{5} x^{3}+4224 B a \,b^{5} d^{3} e^{4} x^{3}-896 B \,b^{6} d^{4} e^{3} x^{3}-10395 A \,a^{4} b^{2} e^{7} x^{2}+27720 A \,a^{3} b^{3} d \,e^{6} x^{2}-33264 A \,a^{2} b^{4} d^{2} e^{5} x^{2}+19008 A a \,b^{5} d^{3} e^{4} x^{2}-4224 A \,b^{6} d^{4} e^{3} x^{2}-4158 B \,a^{5} b \,e^{7} x^{2}+20790 B \,a^{4} b^{2} d \,e^{6} x^{2}-44352 B \,a^{3} b^{3} d^{2} e^{5} x^{2}+47520 B \,a^{2} b^{4} d^{3} e^{4} x^{2}-25344 B a \,b^{5} d^{4} e^{3} x^{2}+5376 B \,b^{6} d^{5} e^{2} x^{2}+4158 A \,a^{5} b \,e^{7} x -41580 A \,a^{4} b^{2} d \,e^{6} x +110880 A \,a^{3} b^{3} d^{2} e^{5} x -133056 A \,a^{2} b^{4} d^{3} e^{4} x +76032 A a \,b^{5} d^{4} e^{3} x -16896 A \,b^{6} d^{5} e^{2} x +693 B \,a^{6} e^{7} x -16632 B \,a^{5} b d \,e^{6} x +83160 B \,a^{4} b^{2} d^{2} e^{5} x -177408 B \,a^{3} b^{3} d^{3} e^{4} x +190080 B \,a^{2} b^{4} d^{4} e^{3} x -101376 B a \,b^{5} d^{5} e^{2} x +21504 B \,b^{6} d^{6} e x +231 A \,a^{6} e^{7}+2772 A \,a^{5} b d \,e^{6}-27720 A \,a^{4} b^{2} d^{2} e^{5}+73920 A \,a^{3} b^{3} d^{3} e^{4}-88704 A \,a^{2} b^{4} d^{4} e^{3}+50688 A a \,b^{5} d^{5} e^{2}-11264 A \,b^{6} d^{6} e +462 B \,a^{6} d \,e^{6}-11088 B \,a^{5} b \,d^{2} e^{5}+55440 B \,a^{4} b^{2} d^{3} e^{4}-118272 B \,a^{3} b^{3} d^{4} e^{3}+126720 B \,a^{2} b^{4} d^{5} e^{2}-67584 B a \,b^{5} d^{6} e +14336 B \,b^{6} d^{7}\right )}{693 \left (e x +d \right )^{\frac {3}{2}} e^{8}} \end {gather*}

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)^3/(e*x+d)^(5/2),x)

[Out]

-2/693*(-63*B*b^6*e^7*x^7-77*A*b^6*e^7*x^6-462*B*a*b^5*e^7*x^6+98*B*b^6*d*e^6*x^6-594*A*a*b^5*e^7*x^5+132*A*b^
6*d*e^6*x^5-1485*B*a^2*b^4*e^7*x^5+792*B*a*b^5*d*e^6*x^5-168*B*b^6*d^2*e^5*x^5-2079*A*a^2*b^4*e^7*x^4+1188*A*a
*b^5*d*e^6*x^4-264*A*b^6*d^2*e^5*x^4-2772*B*a^3*b^3*e^7*x^4+2970*B*a^2*b^4*d*e^6*x^4-1584*B*a*b^5*d^2*e^5*x^4+
336*B*b^6*d^3*e^4*x^4-4620*A*a^3*b^3*e^7*x^3+5544*A*a^2*b^4*d*e^6*x^3-3168*A*a*b^5*d^2*e^5*x^3+704*A*b^6*d^3*e
^4*x^3-3465*B*a^4*b^2*e^7*x^3+7392*B*a^3*b^3*d*e^6*x^3-7920*B*a^2*b^4*d^2*e^5*x^3+4224*B*a*b^5*d^3*e^4*x^3-896
*B*b^6*d^4*e^3*x^3-10395*A*a^4*b^2*e^7*x^2+27720*A*a^3*b^3*d*e^6*x^2-33264*A*a^2*b^4*d^2*e^5*x^2+19008*A*a*b^5
*d^3*e^4*x^2-4224*A*b^6*d^4*e^3*x^2-4158*B*a^5*b*e^7*x^2+20790*B*a^4*b^2*d*e^6*x^2-44352*B*a^3*b^3*d^2*e^5*x^2
+47520*B*a^2*b^4*d^3*e^4*x^2-25344*B*a*b^5*d^4*e^3*x^2+5376*B*b^6*d^5*e^2*x^2+4158*A*a^5*b*e^7*x-41580*A*a^4*b
^2*d*e^6*x+110880*A*a^3*b^3*d^2*e^5*x-133056*A*a^2*b^4*d^3*e^4*x+76032*A*a*b^5*d^4*e^3*x-16896*A*b^6*d^5*e^2*x
+693*B*a^6*e^7*x-16632*B*a^5*b*d*e^6*x+83160*B*a^4*b^2*d^2*e^5*x-177408*B*a^3*b^3*d^3*e^4*x+190080*B*a^2*b^4*d
^4*e^3*x-101376*B*a*b^5*d^5*e^2*x+21504*B*b^6*d^6*e*x+231*A*a^6*e^7+2772*A*a^5*b*d*e^6-27720*A*a^4*b^2*d^2*e^5
+73920*A*a^3*b^3*d^3*e^4-88704*A*a^2*b^4*d^4*e^3+50688*A*a*b^5*d^5*e^2-11264*A*b^6*d^6*e+462*B*a^6*d*e^6-11088
*B*a^5*b*d^2*e^5+55440*B*a^4*b^2*d^3*e^4-118272*B*a^3*b^3*d^4*e^3+126720*B*a^2*b^4*d^5*e^2-67584*B*a*b^5*d^6*e
+14336*B*b^6*d^7)/(e*x+d)^(3/2)/e^8

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maxima [B]  time = 0.53, size = 773, normalized size = 2.56 \begin {gather*} \frac {2 \, {\left (\frac {63 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{6} - 77 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 297 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (7 \, B b^{6} d^{3} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (7 \, B b^{6} d^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 2079 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} \sqrt {e x + d}}{e^{7}} + \frac {231 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 3 \, {\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{7}}\right )}}{693 \, e} \end {gather*}

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)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/693*((63*(e*x + d)^(11/2)*B*b^6 - 77*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(9/2) + 297*(7*B*b^6*d^2
- 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(7/2) - 693*(7*B*b^6*d^3 - 3*(6*B*a*b^5
+ A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*x + d)^(5/2) + 1155*
(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^
4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^(3/2) - 2079*(7*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e
+ 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3
)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*sqrt(e*x + d))/e^7 + 231*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)
*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 - 3*(7*B*b^6*d^6 - 6*(6*B*a*b
^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*a^
4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*(e*x + d))/((e*x +
d)^(3/2)*e^7))/e

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mupad [B]  time = 1.97, size = 569, normalized size = 1.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{9\,e^8}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^6\,e^6-24\,B\,a^5\,b\,d\,e^5+12\,A\,a^5\,b\,e^6+90\,B\,a^4\,b^2\,d^2\,e^4-60\,A\,a^4\,b^2\,d\,e^5-160\,B\,a^3\,b^3\,d^3\,e^3+120\,A\,a^3\,b^3\,d^2\,e^4+150\,B\,a^2\,b^4\,d^4\,e^2-120\,A\,a^2\,b^4\,d^3\,e^3-72\,B\,a\,b^5\,d^5\,e+60\,A\,a\,b^5\,d^4\,e^2+14\,B\,b^6\,d^6-12\,A\,b^6\,d^5\,e\right )+\frac {2\,A\,a^6\,e^7}{3}-\frac {2\,B\,b^6\,d^7}{3}+\frac {2\,A\,b^6\,d^6\,e}{3}-\frac {2\,B\,a^6\,d\,e^6}{3}-4\,A\,a\,b^5\,d^5\,e^2+4\,B\,a^5\,b\,d^2\,e^5+10\,A\,a^2\,b^4\,d^4\,e^3-\frac {40\,A\,a^3\,b^3\,d^3\,e^4}{3}+10\,A\,a^4\,b^2\,d^2\,e^5-10\,B\,a^2\,b^4\,d^5\,e^2+\frac {40\,B\,a^3\,b^3\,d^4\,e^3}{3}-10\,B\,a^4\,b^2\,d^3\,e^4-4\,A\,a^5\,b\,d\,e^6+4\,B\,a\,b^5\,d^6\,e}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8}+\frac {2\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{e^8} \end {gather*}

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)^3)/(d + e*x)^(5/2),x)

[Out]

((d + e*x)^(9/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(9*e^8) - ((d + e*x)*(2*B*a^6*e^6 + 14*B*b^6*d^6 + 1
2*A*a^5*b*e^6 - 12*A*b^6*d^5*e + 60*A*a*b^5*d^4*e^2 - 60*A*a^4*b^2*d*e^5 - 120*A*a^2*b^4*d^3*e^3 + 120*A*a^3*b
^3*d^2*e^4 + 150*B*a^2*b^4*d^4*e^2 - 160*B*a^3*b^3*d^3*e^3 + 90*B*a^4*b^2*d^2*e^4 - 72*B*a*b^5*d^5*e - 24*B*a^
5*b*d*e^5) + (2*A*a^6*e^7)/3 - (2*B*b^6*d^7)/3 + (2*A*b^6*d^6*e)/3 - (2*B*a^6*d*e^6)/3 - 4*A*a*b^5*d^5*e^2 + 4
*B*a^5*b*d^2*e^5 + 10*A*a^2*b^4*d^4*e^3 - (40*A*a^3*b^3*d^3*e^4)/3 + 10*A*a^4*b^2*d^2*e^5 - 10*B*a^2*b^4*d^5*e
^2 + (40*B*a^3*b^3*d^4*e^3)/3 - 10*B*a^4*b^2*d^3*e^4 - 4*A*a^5*b*d*e^6 + 4*B*a*b^5*d^6*e)/(e^8*(d + e*x)^(3/2)
) + (2*B*b^6*(d + e*x)^(11/2))/(11*e^8) + (6*b*(a*e - b*d)^4*(d + e*x)^(1/2)*(5*A*b*e + 2*B*a*e - 7*B*b*d))/e^
8 + (6*b^4*(a*e - b*d)*(d + e*x)^(7/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(7*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x
)^(3/2)*(4*A*b*e + 3*B*a*e - 7*B*b*d))/(3*e^8) + (2*b^3*(a*e - b*d)^2*(d + e*x)^(5/2)*(3*A*b*e + 4*B*a*e - 7*B
*b*d))/e^8

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sympy [F(-1)]  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)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(5/2),x)

[Out]

Timed out

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