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

Optimal. Leaf size=306 \[ -\frac {2 b^5 (d+e x)^{13/2} (-6 a B e-A b e+7 b B d)}{13 e^8}+\frac {6 b^4 (d+e x)^{11/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{11 e^8}-\frac {10 b^3 (d+e x)^{9/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8}+\frac {10 b^2 (d+e x)^{7/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{7 e^8}-\frac {6 b (d+e x)^{5/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{5 e^8}+\frac {2 (d+e x)^{3/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^6 (B d-A e)}{e^8}+\frac {2 b^6 B (d+e x)^{15/2}}{15 e^8} \]

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Rubi [A]  time = 0.15, antiderivative size = 306, 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)^{13/2} (-6 a B e-A b e+7 b B d)}{13 e^8}+\frac {6 b^4 (d+e x)^{11/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{11 e^8}-\frac {10 b^3 (d+e x)^{9/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8}+\frac {10 b^2 (d+e x)^{7/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{7 e^8}-\frac {6 b (d+e x)^{5/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{5 e^8}+\frac {2 (d+e x)^{3/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^6 (B d-A e)}{e^8}+\frac {2 b^6 B (d+e x)^{15/2}}{15 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^6*(B*d - A*e)*Sqrt[d + e*x])/e^8 + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^(3/2
))/(3*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(5/2))/(5*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)^(7/2))/(7*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*
(d + e*x)^(9/2))/(9*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (2*b^
5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^6*B*(d + e*x)^(15/2))/(15*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}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (-B d+A e)}{e^7 \sqrt {d+e x}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e) \sqrt {d+e x}}{e^7}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e) (d+e x)^{3/2}}{e^7}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) (d+e x)^{5/2}}{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)^{7/2}}{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)^{9/2}}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{11/2}}{e^7}+\frac {b^6 B (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6 (B d-A e) \sqrt {d+e x}}{e^8}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{3/2}}{3 e^8}-\frac {6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{5/2}}{5 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)^{7/2}}{7 e^8}-\frac {10 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{9/2}}{9 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)^{11/2}}{11 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{13/2}}{13 e^8}+\frac {2 b^6 B (d+e x)^{15/2}}{15 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 259, normalized size = 0.85 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-3465 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+12285 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-25025 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+32175 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-27027 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+15015 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-45045 (b d-a e)^6 (B d-A e)+3003 b^6 B (d+e x)^7\right )}{45045 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-45045*(b*d - a*e)^6*(B*d - A*e) + 15015*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)
- 27027*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 32175*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*
e - 3*a*B*e)*(d + e*x)^3 - 25025*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 12285*b^4*(b*d
- a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5 - 3465*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 3003*b^6
*B*(d + e*x)^7))/(45045*e^8)

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IntegrateAlgebraic [B]  time = 0.41, size = 1310, normalized size = 4.28 \begin {gather*} -\frac {2 \left (-3003 b^6 B (d+e x)^{15/2}+24255 b^6 B d (d+e x)^{13/2}-3465 A b^6 e (d+e x)^{13/2}-20790 a b^5 B e (d+e x)^{13/2}-85995 b^6 B d^2 (d+e x)^{11/2}-24570 a A b^5 e^2 (d+e x)^{11/2}-61425 a^2 b^4 B e^2 (d+e x)^{11/2}+24570 A b^6 d e (d+e x)^{11/2}+147420 a b^5 B d e (d+e x)^{11/2}+175175 b^6 B d^3 (d+e x)^{9/2}-75075 a^2 A b^4 e^3 (d+e x)^{9/2}-100100 a^3 b^3 B e^3 (d+e x)^{9/2}+150150 a A b^5 d e^2 (d+e x)^{9/2}+375375 a^2 b^4 B d e^2 (d+e x)^{9/2}-75075 A b^6 d^2 e (d+e x)^{9/2}-450450 a b^5 B d^2 e (d+e x)^{9/2}-225225 b^6 B d^4 (d+e x)^{7/2}-128700 a^3 A b^3 e^4 (d+e x)^{7/2}-96525 a^4 b^2 B e^4 (d+e x)^{7/2}+386100 a^2 A b^4 d e^3 (d+e x)^{7/2}+514800 a^3 b^3 B d e^3 (d+e x)^{7/2}-386100 a A b^5 d^2 e^2 (d+e x)^{7/2}-965250 a^2 b^4 B d^2 e^2 (d+e x)^{7/2}+128700 A b^6 d^3 e (d+e x)^{7/2}+772200 a b^5 B d^3 e (d+e x)^{7/2}+189189 b^6 B d^5 (d+e x)^{5/2}-135135 a^4 A b^2 e^5 (d+e x)^{5/2}-54054 a^5 b B e^5 (d+e x)^{5/2}+540540 a^3 A b^3 d e^4 (d+e x)^{5/2}+405405 a^4 b^2 B d e^4 (d+e x)^{5/2}-810810 a^2 A b^4 d^2 e^3 (d+e x)^{5/2}-1081080 a^3 b^3 B d^2 e^3 (d+e x)^{5/2}+540540 a A b^5 d^3 e^2 (d+e x)^{5/2}+1351350 a^2 b^4 B d^3 e^2 (d+e x)^{5/2}-135135 A b^6 d^4 e (d+e x)^{5/2}-810810 a b^5 B d^4 e (d+e x)^{5/2}-105105 b^6 B d^6 (d+e x)^{3/2}-90090 a^5 A b e^6 (d+e x)^{3/2}-15015 a^6 B e^6 (d+e x)^{3/2}+450450 a^4 A b^2 d e^5 (d+e x)^{3/2}+180180 a^5 b B d e^5 (d+e x)^{3/2}-900900 a^3 A b^3 d^2 e^4 (d+e x)^{3/2}-675675 a^4 b^2 B d^2 e^4 (d+e x)^{3/2}+900900 a^2 A b^4 d^3 e^3 (d+e x)^{3/2}+1201200 a^3 b^3 B d^3 e^3 (d+e x)^{3/2}-450450 a A b^5 d^4 e^2 (d+e x)^{3/2}-1126125 a^2 b^4 B d^4 e^2 (d+e x)^{3/2}+90090 A b^6 d^5 e (d+e x)^{3/2}+540540 a b^5 B d^5 e (d+e x)^{3/2}+45045 b^6 B d^7 \sqrt {d+e x}-45045 a^6 A e^7 \sqrt {d+e x}+270270 a^5 A b d e^6 \sqrt {d+e x}+45045 a^6 B d e^6 \sqrt {d+e x}-675675 a^4 A b^2 d^2 e^5 \sqrt {d+e x}-270270 a^5 b B d^2 e^5 \sqrt {d+e x}+900900 a^3 A b^3 d^3 e^4 \sqrt {d+e x}+675675 a^4 b^2 B d^3 e^4 \sqrt {d+e x}-675675 a^2 A b^4 d^4 e^3 \sqrt {d+e x}-900900 a^3 b^3 B d^4 e^3 \sqrt {d+e x}+270270 a A b^5 d^5 e^2 \sqrt {d+e x}+675675 a^2 b^4 B d^5 e^2 \sqrt {d+e x}-45045 A b^6 d^6 e \sqrt {d+e x}-270270 a b^5 B d^6 e \sqrt {d+e x}\right )}{45045 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(45045*b^6*B*d^7*Sqrt[d + e*x] - 45045*A*b^6*d^6*e*Sqrt[d + e*x] - 270270*a*b^5*B*d^6*e*Sqrt[d + e*x] + 27
0270*a*A*b^5*d^5*e^2*Sqrt[d + e*x] + 675675*a^2*b^4*B*d^5*e^2*Sqrt[d + e*x] - 675675*a^2*A*b^4*d^4*e^3*Sqrt[d
+ e*x] - 900900*a^3*b^3*B*d^4*e^3*Sqrt[d + e*x] + 900900*a^3*A*b^3*d^3*e^4*Sqrt[d + e*x] + 675675*a^4*b^2*B*d^
3*e^4*Sqrt[d + e*x] - 675675*a^4*A*b^2*d^2*e^5*Sqrt[d + e*x] - 270270*a^5*b*B*d^2*e^5*Sqrt[d + e*x] + 270270*a
^5*A*b*d*e^6*Sqrt[d + e*x] + 45045*a^6*B*d*e^6*Sqrt[d + e*x] - 45045*a^6*A*e^7*Sqrt[d + e*x] - 105105*b^6*B*d^
6*(d + e*x)^(3/2) + 90090*A*b^6*d^5*e*(d + e*x)^(3/2) + 540540*a*b^5*B*d^5*e*(d + e*x)^(3/2) - 450450*a*A*b^5*
d^4*e^2*(d + e*x)^(3/2) - 1126125*a^2*b^4*B*d^4*e^2*(d + e*x)^(3/2) + 900900*a^2*A*b^4*d^3*e^3*(d + e*x)^(3/2)
+ 1201200*a^3*b^3*B*d^3*e^3*(d + e*x)^(3/2) - 900900*a^3*A*b^3*d^2*e^4*(d + e*x)^(3/2) - 675675*a^4*b^2*B*d^2
*e^4*(d + e*x)^(3/2) + 450450*a^4*A*b^2*d*e^5*(d + e*x)^(3/2) + 180180*a^5*b*B*d*e^5*(d + e*x)^(3/2) - 90090*a
^5*A*b*e^6*(d + e*x)^(3/2) - 15015*a^6*B*e^6*(d + e*x)^(3/2) + 189189*b^6*B*d^5*(d + e*x)^(5/2) - 135135*A*b^6
*d^4*e*(d + e*x)^(5/2) - 810810*a*b^5*B*d^4*e*(d + e*x)^(5/2) + 540540*a*A*b^5*d^3*e^2*(d + e*x)^(5/2) + 13513
50*a^2*b^4*B*d^3*e^2*(d + e*x)^(5/2) - 810810*a^2*A*b^4*d^2*e^3*(d + e*x)^(5/2) - 1081080*a^3*b^3*B*d^2*e^3*(d
+ e*x)^(5/2) + 540540*a^3*A*b^3*d*e^4*(d + e*x)^(5/2) + 405405*a^4*b^2*B*d*e^4*(d + e*x)^(5/2) - 135135*a^4*A
*b^2*e^5*(d + e*x)^(5/2) - 54054*a^5*b*B*e^5*(d + e*x)^(5/2) - 225225*b^6*B*d^4*(d + e*x)^(7/2) + 128700*A*b^6
*d^3*e*(d + e*x)^(7/2) + 772200*a*b^5*B*d^3*e*(d + e*x)^(7/2) - 386100*a*A*b^5*d^2*e^2*(d + e*x)^(7/2) - 96525
0*a^2*b^4*B*d^2*e^2*(d + e*x)^(7/2) + 386100*a^2*A*b^4*d*e^3*(d + e*x)^(7/2) + 514800*a^3*b^3*B*d*e^3*(d + e*x
)^(7/2) - 128700*a^3*A*b^3*e^4*(d + e*x)^(7/2) - 96525*a^4*b^2*B*e^4*(d + e*x)^(7/2) + 175175*b^6*B*d^3*(d + e
*x)^(9/2) - 75075*A*b^6*d^2*e*(d + e*x)^(9/2) - 450450*a*b^5*B*d^2*e*(d + e*x)^(9/2) + 150150*a*A*b^5*d*e^2*(d
+ e*x)^(9/2) + 375375*a^2*b^4*B*d*e^2*(d + e*x)^(9/2) - 75075*a^2*A*b^4*e^3*(d + e*x)^(9/2) - 100100*a^3*b^3*
B*e^3*(d + e*x)^(9/2) - 85995*b^6*B*d^2*(d + e*x)^(11/2) + 24570*A*b^6*d*e*(d + e*x)^(11/2) + 147420*a*b^5*B*d
*e*(d + e*x)^(11/2) - 24570*a*A*b^5*e^2*(d + e*x)^(11/2) - 61425*a^2*b^4*B*e^2*(d + e*x)^(11/2) + 24255*b^6*B*
d*(d + e*x)^(13/2) - 3465*A*b^6*e*(d + e*x)^(13/2) - 20790*a*b^5*B*e*(d + e*x)^(13/2) - 3003*b^6*B*(d + e*x)^(
15/2)))/(45045*e^8)

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fricas [B]  time = 0.48, size = 769, normalized size = 2.51 \begin {gather*} \frac {2 \, {\left (3003 \, B b^{6} e^{7} x^{7} - 14336 \, B b^{6} d^{7} + 45045 \, A a^{6} e^{7} + 15360 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e - 49920 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 91520 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} - 102960 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 72072 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - 30030 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 231 \, {\left (14 \, B b^{6} d e^{6} - 15 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 63 \, {\left (56 \, B b^{6} d^{2} e^{5} - 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 195 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 35 \, {\left (112 \, B b^{6} d^{3} e^{4} - 120 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 390 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 715 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 5 \, {\left (896 \, B b^{6} d^{4} e^{3} - 960 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 3120 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 5720 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 6435 \, {\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} - 1920 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 6240 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 11440 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 12870 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 9009 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + {\left (7168 \, B b^{6} d^{6} e - 7680 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 24960 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 45760 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 51480 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 36036 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 15015 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{8}} \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)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^6*e^7*x^7 - 14336*B*b^6*d^7 + 45045*A*a^6*e^7 + 15360*(6*B*a*b^5 + A*b^6)*d^6*e - 49920*(5*B
*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 91520*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 102960*(3*B*a^4*b^2 + 4*A*a^3*b^3)
*d^3*e^4 + 72072*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 30030*(B*a^6 + 6*A*a^5*b)*d*e^6 - 231*(14*B*b^6*d*e^6 - 1
5*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 63*(56*B*b^6*d^2*e^5 - 60*(6*B*a*b^5 + A*b^6)*d*e^6 + 195*(5*B*a^2*b^4 + 2*A*
a*b^5)*e^7)*x^5 - 35*(112*B*b^6*d^3*e^4 - 120*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 390*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^
6 - 715*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 5*(896*B*b^6*d^4*e^3 - 960*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3120*(
5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 5720*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 6435*(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 - 1920*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6240*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 -
11440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 12870*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 9009*(2*B*a^5*b + 5*A*a^
4*b^2)*e^7)*x^2 + (7168*B*b^6*d^6*e - 7680*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 24960*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e
^3 - 45760*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 51480*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 36036*(2*B*a^5*b
+ 5*A*a^4*b^2)*d*e^6 + 15015*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/e^8

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giac [B]  time = 0.24, size = 895, normalized size = 2.92

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)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^6*e^(-1) + 90090*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d
)*A*a^5*b*e^(-1) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^5*b*e^(-2) + 45
045*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*b^2*e^(-2) + 19305*(5*(x*e + d)^(7
/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^4*b^2*e^(-3) + 25740*(5*(x*e +
d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^3*b^3*e^(-3) + 2860*(35*
(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d
)*d^4)*B*a^3*b^3*e^(-4) + 2145*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*
e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b^4*e^(-4) + 975*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d
+ 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2
*b^4*e^(-5) + 390*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2
)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a*b^5*e^(-5) + 90*(231*(x*e + d)^(13/2) - 1638*(x*
e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e +
d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a*b^5*e^(-6) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d +
5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 30
03*sqrt(x*e + d)*d^6)*A*b^6*e^(-6) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2
)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3
/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b^6*e^(-7) + 45045*sqrt(x*e + d)*A*a^6)*e^(-1)

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maple [B]  time = 0.06, size = 913, normalized size = 2.98 \begin {gather*} \frac {2 \left (3003 B \,b^{6} x^{7} e^{7}+3465 A \,b^{6} e^{7} x^{6}+20790 B a \,b^{5} e^{7} x^{6}-3234 B \,b^{6} d \,e^{6} x^{6}+24570 A a \,b^{5} e^{7} x^{5}-3780 A \,b^{6} d \,e^{6} x^{5}+61425 B \,a^{2} b^{4} e^{7} x^{5}-22680 B a \,b^{5} d \,e^{6} x^{5}+3528 B \,b^{6} d^{2} e^{5} x^{5}+75075 A \,a^{2} b^{4} e^{7} x^{4}-27300 A a \,b^{5} d \,e^{6} x^{4}+4200 A \,b^{6} d^{2} e^{5} x^{4}+100100 B \,a^{3} b^{3} e^{7} x^{4}-68250 B \,a^{2} b^{4} d \,e^{6} x^{4}+25200 B a \,b^{5} d^{2} e^{5} x^{4}-3920 B \,b^{6} d^{3} e^{4} x^{4}+128700 A \,a^{3} b^{3} e^{7} x^{3}-85800 A \,a^{2} b^{4} d \,e^{6} x^{3}+31200 A a \,b^{5} d^{2} e^{5} x^{3}-4800 A \,b^{6} d^{3} e^{4} x^{3}+96525 B \,a^{4} b^{2} e^{7} x^{3}-114400 B \,a^{3} b^{3} d \,e^{6} x^{3}+78000 B \,a^{2} b^{4} d^{2} e^{5} x^{3}-28800 B a \,b^{5} d^{3} e^{4} x^{3}+4480 B \,b^{6} d^{4} e^{3} x^{3}+135135 A \,a^{4} b^{2} e^{7} x^{2}-154440 A \,a^{3} b^{3} d \,e^{6} x^{2}+102960 A \,a^{2} b^{4} d^{2} e^{5} x^{2}-37440 A a \,b^{5} d^{3} e^{4} x^{2}+5760 A \,b^{6} d^{4} e^{3} x^{2}+54054 B \,a^{5} b \,e^{7} x^{2}-115830 B \,a^{4} b^{2} d \,e^{6} x^{2}+137280 B \,a^{3} b^{3} d^{2} e^{5} x^{2}-93600 B \,a^{2} b^{4} d^{3} e^{4} x^{2}+34560 B a \,b^{5} d^{4} e^{3} x^{2}-5376 B \,b^{6} d^{5} e^{2} x^{2}+90090 A \,a^{5} b \,e^{7} x -180180 A \,a^{4} b^{2} d \,e^{6} x +205920 A \,a^{3} b^{3} d^{2} e^{5} x -137280 A \,a^{2} b^{4} d^{3} e^{4} x +49920 A a \,b^{5} d^{4} e^{3} x -7680 A \,b^{6} d^{5} e^{2} x +15015 B \,a^{6} e^{7} x -72072 B \,a^{5} b d \,e^{6} x +154440 B \,a^{4} b^{2} d^{2} e^{5} x -183040 B \,a^{3} b^{3} d^{3} e^{4} x +124800 B \,a^{2} b^{4} d^{4} e^{3} x -46080 B a \,b^{5} d^{5} e^{2} x +7168 B \,b^{6} d^{6} e x +45045 A \,a^{6} e^{7}-180180 A \,a^{5} b d \,e^{6}+360360 A \,a^{4} b^{2} d^{2} e^{5}-411840 A \,a^{3} b^{3} d^{3} e^{4}+274560 A \,a^{2} b^{4} d^{4} e^{3}-99840 A a \,b^{5} d^{5} e^{2}+15360 A \,b^{6} d^{6} e -30030 B \,a^{6} d \,e^{6}+144144 B \,a^{5} b \,d^{2} e^{5}-308880 B \,a^{4} b^{2} d^{3} e^{4}+366080 B \,a^{3} b^{3} d^{4} e^{3}-249600 B \,a^{2} b^{4} d^{5} e^{2}+92160 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right ) \sqrt {e x +d}}{45045 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)^(1/2),x)

[Out]

2/45045*(3003*B*b^6*e^7*x^7+3465*A*b^6*e^7*x^6+20790*B*a*b^5*e^7*x^6-3234*B*b^6*d*e^6*x^6+24570*A*a*b^5*e^7*x^
5-3780*A*b^6*d*e^6*x^5+61425*B*a^2*b^4*e^7*x^5-22680*B*a*b^5*d*e^6*x^5+3528*B*b^6*d^2*e^5*x^5+75075*A*a^2*b^4*
e^7*x^4-27300*A*a*b^5*d*e^6*x^4+4200*A*b^6*d^2*e^5*x^4+100100*B*a^3*b^3*e^7*x^4-68250*B*a^2*b^4*d*e^6*x^4+2520
0*B*a*b^5*d^2*e^5*x^4-3920*B*b^6*d^3*e^4*x^4+128700*A*a^3*b^3*e^7*x^3-85800*A*a^2*b^4*d*e^6*x^3+31200*A*a*b^5*
d^2*e^5*x^3-4800*A*b^6*d^3*e^4*x^3+96525*B*a^4*b^2*e^7*x^3-114400*B*a^3*b^3*d*e^6*x^3+78000*B*a^2*b^4*d^2*e^5*
x^3-28800*B*a*b^5*d^3*e^4*x^3+4480*B*b^6*d^4*e^3*x^3+135135*A*a^4*b^2*e^7*x^2-154440*A*a^3*b^3*d*e^6*x^2+10296
0*A*a^2*b^4*d^2*e^5*x^2-37440*A*a*b^5*d^3*e^4*x^2+5760*A*b^6*d^4*e^3*x^2+54054*B*a^5*b*e^7*x^2-115830*B*a^4*b^
2*d*e^6*x^2+137280*B*a^3*b^3*d^2*e^5*x^2-93600*B*a^2*b^4*d^3*e^4*x^2+34560*B*a*b^5*d^4*e^3*x^2-5376*B*b^6*d^5*
e^2*x^2+90090*A*a^5*b*e^7*x-180180*A*a^4*b^2*d*e^6*x+205920*A*a^3*b^3*d^2*e^5*x-137280*A*a^2*b^4*d^3*e^4*x+499
20*A*a*b^5*d^4*e^3*x-7680*A*b^6*d^5*e^2*x+15015*B*a^6*e^7*x-72072*B*a^5*b*d*e^6*x+154440*B*a^4*b^2*d^2*e^5*x-1
83040*B*a^3*b^3*d^3*e^4*x+124800*B*a^2*b^4*d^4*e^3*x-46080*B*a*b^5*d^5*e^2*x+7168*B*b^6*d^6*e*x+45045*A*a^6*e^
7-180180*A*a^5*b*d*e^6+360360*A*a^4*b^2*d^2*e^5-411840*A*a^3*b^3*d^3*e^4+274560*A*a^2*b^4*d^4*e^3-99840*A*a*b^
5*d^5*e^2+15360*A*b^6*d^6*e-30030*B*a^6*d*e^6+144144*B*a^5*b*d^2*e^5-308880*B*a^4*b^2*d^3*e^4+366080*B*a^3*b^3
*d^4*e^3-249600*B*a^2*b^4*d^5*e^2+92160*B*a*b^5*d^6*e-14336*B*b^6*d^7)*(e*x+d)^(1/2)/e^8

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maxima [B]  time = 0.61, size = 767, normalized size = 2.51 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{6} - 3465 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\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 {11}{2}} - 25025 \, {\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 {9}{2}} + 32175 \, {\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 {7}{2}} - 27027 \, {\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 )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\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 )}^{\frac {3}{2}} - 45045 \, {\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}\right )} \sqrt {e x + d}\right )}}{45045 \, e^{8}} \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)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*b^6 - 3465*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(13/2) + 12285*(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)^(11/2) - 25025*(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)^(9
/2) + 32175*(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)^(7/2) - 27027*(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)*(e*x + d)^(5/2) + 15015*(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)^(3/2) - 45045*(
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)*sqrt(e*x + d))/e^8

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mupad [B]  time = 0.08, size = 279, normalized size = 0.91 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{13\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{3\,e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^8}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{5\,e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{11\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{9\,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)^(1/2),x)

[Out]

((d + e*x)^(13/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(13*e^8) + (2*(a*e - b*d)^5*(d + e*x)^(3/2)*(6*A*b*
e + B*a*e - 7*B*b*d))/(3*e^8) + (2*B*b^6*(d + e*x)^(15/2))/(15*e^8) + (2*(A*e - B*d)*(a*e - b*d)^6*(d + e*x)^(
1/2))/e^8 + (6*b*(a*e - b*d)^4*(d + e*x)^(5/2)*(5*A*b*e + 2*B*a*e - 7*B*b*d))/(5*e^8) + (6*b^4*(a*e - b*d)*(d
+ e*x)^(11/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(11*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(7/2)*(4*A*b*e + 3*B*a
*e - 7*B*b*d))/(7*e^8) + (10*b^3*(a*e - b*d)^2*(d + e*x)^(9/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(9*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)**(1/2),x)

[Out]

Timed out

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