3.15.18 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=425 \[ \frac {2 (d+e x)^{7/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^8}+\frac {6 c^2 (d+e x)^{11/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^8}-\frac {10 c (d+e x)^{9/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac {6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {14 c^3 (d+e x)^{13/2} (2 c d-b e)}{13 e^8}+\frac {4 c^4 (d+e x)^{15/2}}{15 e^8} \]

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Rubi [A]  time = 0.23, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {771} \begin {gather*} \frac {2 (d+e x)^{7/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^8}+\frac {6 c^2 (d+e x)^{11/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^8}-\frac {10 c (d+e x)^{9/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac {6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {14 c^3 (d+e x)^{13/2} (2 c d-b e)}{13 e^8}+\frac {4 c^4 (d+e x)^{15/2}}{15 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x])/e^8 + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*
b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(3/2))/(3*e^8) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2
+ b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(5/2))/(5*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*
a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(7/2))/(7*e^8)
- (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(9/2))/(9*e^8) + (6*c^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(11/2))/(11*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*
e^8) + (4*c^4*(d + e*x)^(15/2))/(15*e^8)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 \sqrt {d+e x}}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \sqrt {d+e x}}{e^7}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{3/2}}{e^7}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{5/2}}{e^7}+\frac {5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{7/2}}{e^7}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{9/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{11/2}}{e^7}+\frac {2 c^4 (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}{e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{3/2}}{3 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5/2}}{5 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{9 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{11/2}}{11 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{13/2}}{13 e^8}+\frac {4 c^4 (d+e x)^{15/2}}{15 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.62, size = 599, normalized size = 1.41 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-39 c^2 e^2 \left (198 a^2 e^2 \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-55 a b e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+15 b^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )+143 c e^3 \left (210 a^3 e^3 (e x-2 d)+189 a^2 b e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+108 a b^2 e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+5 b^3 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+1287 b e^4 \left (35 a^3 e^3+35 a^2 b e^2 (e x-2 d)+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )+15 c^3 e \left (26 a e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+7 b \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )-14 c^4 \left (2048 d^7-1024 d^6 e x+768 d^5 e^2 x^2-640 d^4 e^3 x^3+560 d^3 e^4 x^4-504 d^2 e^5 x^5+462 d e^6 x^6-429 e^7 x^7\right )\right )}{45045 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-14*c^4*(2048*d^7 - 1024*d^6*e*x + 768*d^5*e^2*x^2 - 640*d^4*e^3*x^3 + 560*d^3*e^4*x^4 - 504
*d^2*e^5*x^5 + 462*d*e^6*x^6 - 429*e^7*x^7) + 1287*b*e^4*(35*a^3*e^3 + 35*a^2*b*e^2*(-2*d + e*x) + 7*a*b^2*e*(
8*d^2 - 4*d*e*x + 3*e^2*x^2) + b^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)) + 143*c*e^3*(210*a^3*e^3*(
-2*d + e*x) + 189*a^2*b*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 108*a*b^2*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5
*e^3*x^3) + 5*b^3*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)) - 39*c^2*e^2*(198*a^2*e
^2*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3) - 55*a*b*e*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3
*x^3 + 35*e^4*x^4) + 15*b^2*(256*d^5 - 128*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x
^5)) + 15*c^3*e*(26*a*e*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5)
+ 7*b*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6
*x^6))))/(45045*e^8)

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IntegrateAlgebraic [B]  time = 0.37, size = 1186, normalized size = 2.79 \begin {gather*} -\frac {2 \left (-6006 c^4 (d+e x)^{15/2}+48510 c^4 d (d+e x)^{13/2}-24255 b c^3 e (d+e x)^{13/2}-171990 c^4 d^2 (d+e x)^{11/2}-24570 a c^3 e^2 (d+e x)^{11/2}-36855 b^2 c^2 e^2 (d+e x)^{11/2}+171990 b c^3 d e (d+e x)^{11/2}+350350 c^4 d^3 (d+e x)^{9/2}-75075 a b c^2 e^3 (d+e x)^{9/2}-25025 b^3 c e^3 (d+e x)^{9/2}+150150 a c^3 d e^2 (d+e x)^{9/2}+225225 b^2 c^2 d e^2 (d+e x)^{9/2}-525525 b c^3 d^2 e (d+e x)^{9/2}-450450 c^4 d^4 (d+e x)^{7/2}-6435 b^4 e^4 (d+e x)^{7/2}-38610 a^2 c^2 e^4 (d+e x)^{7/2}-77220 a b^2 c e^4 (d+e x)^{7/2}+386100 a b c^2 d e^3 (d+e x)^{7/2}+128700 b^3 c d e^3 (d+e x)^{7/2}-386100 a c^3 d^2 e^2 (d+e x)^{7/2}-579150 b^2 c^2 d^2 e^2 (d+e x)^{7/2}+900900 b c^3 d^3 e (d+e x)^{7/2}+378378 c^4 d^5 (d+e x)^{5/2}-27027 a b^3 e^5 (d+e x)^{5/2}-81081 a^2 b c e^5 (d+e x)^{5/2}+27027 b^4 d e^4 (d+e x)^{5/2}+162162 a^2 c^2 d e^4 (d+e x)^{5/2}+324324 a b^2 c d e^4 (d+e x)^{5/2}-810810 a b c^2 d^2 e^3 (d+e x)^{5/2}-270270 b^3 c d^2 e^3 (d+e x)^{5/2}+540540 a c^3 d^3 e^2 (d+e x)^{5/2}+810810 b^2 c^2 d^3 e^2 (d+e x)^{5/2}-945945 b c^3 d^4 e (d+e x)^{5/2}-210210 c^4 d^6 (d+e x)^{3/2}-45045 a^2 b^2 e^6 (d+e x)^{3/2}-30030 a^3 c e^6 (d+e x)^{3/2}+90090 a b^3 d e^5 (d+e x)^{3/2}+270270 a^2 b c d e^5 (d+e x)^{3/2}-45045 b^4 d^2 e^4 (d+e x)^{3/2}-270270 a^2 c^2 d^2 e^4 (d+e x)^{3/2}-540540 a b^2 c d^2 e^4 (d+e x)^{3/2}+900900 a b c^2 d^3 e^3 (d+e x)^{3/2}+300300 b^3 c d^3 e^3 (d+e x)^{3/2}-450450 a c^3 d^4 e^2 (d+e x)^{3/2}-675675 b^2 c^2 d^4 e^2 (d+e x)^{3/2}+630630 b c^3 d^5 e (d+e x)^{3/2}+90090 c^4 d^7 \sqrt {d+e x}-45045 a^3 b e^7 \sqrt {d+e x}+135135 a^2 b^2 d e^6 \sqrt {d+e x}+90090 a^3 c d e^6 \sqrt {d+e x}-135135 a b^3 d^2 e^5 \sqrt {d+e x}-405405 a^2 b c d^2 e^5 \sqrt {d+e x}+45045 b^4 d^3 e^4 \sqrt {d+e x}+270270 a^2 c^2 d^3 e^4 \sqrt {d+e x}+540540 a b^2 c d^3 e^4 \sqrt {d+e x}-675675 a b c^2 d^4 e^3 \sqrt {d+e x}-225225 b^3 c d^4 e^3 \sqrt {d+e x}+270270 a c^3 d^5 e^2 \sqrt {d+e x}+405405 b^2 c^2 d^5 e^2 \sqrt {d+e x}-315315 b c^3 d^6 e \sqrt {d+e x}\right )}{45045 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(90090*c^4*d^7*Sqrt[d + e*x] - 315315*b*c^3*d^6*e*Sqrt[d + e*x] + 405405*b^2*c^2*d^5*e^2*Sqrt[d + e*x] + 2
70270*a*c^3*d^5*e^2*Sqrt[d + e*x] - 225225*b^3*c*d^4*e^3*Sqrt[d + e*x] - 675675*a*b*c^2*d^4*e^3*Sqrt[d + e*x]
+ 45045*b^4*d^3*e^4*Sqrt[d + e*x] + 540540*a*b^2*c*d^3*e^4*Sqrt[d + e*x] + 270270*a^2*c^2*d^3*e^4*Sqrt[d + e*x
] - 135135*a*b^3*d^2*e^5*Sqrt[d + e*x] - 405405*a^2*b*c*d^2*e^5*Sqrt[d + e*x] + 135135*a^2*b^2*d*e^6*Sqrt[d +
e*x] + 90090*a^3*c*d*e^6*Sqrt[d + e*x] - 45045*a^3*b*e^7*Sqrt[d + e*x] - 210210*c^4*d^6*(d + e*x)^(3/2) + 6306
30*b*c^3*d^5*e*(d + e*x)^(3/2) - 675675*b^2*c^2*d^4*e^2*(d + e*x)^(3/2) - 450450*a*c^3*d^4*e^2*(d + e*x)^(3/2)
+ 300300*b^3*c*d^3*e^3*(d + e*x)^(3/2) + 900900*a*b*c^2*d^3*e^3*(d + e*x)^(3/2) - 45045*b^4*d^2*e^4*(d + e*x)
^(3/2) - 540540*a*b^2*c*d^2*e^4*(d + e*x)^(3/2) - 270270*a^2*c^2*d^2*e^4*(d + e*x)^(3/2) + 90090*a*b^3*d*e^5*(
d + e*x)^(3/2) + 270270*a^2*b*c*d*e^5*(d + e*x)^(3/2) - 45045*a^2*b^2*e^6*(d + e*x)^(3/2) - 30030*a^3*c*e^6*(d
+ e*x)^(3/2) + 378378*c^4*d^5*(d + e*x)^(5/2) - 945945*b*c^3*d^4*e*(d + e*x)^(5/2) + 810810*b^2*c^2*d^3*e^2*(
d + e*x)^(5/2) + 540540*a*c^3*d^3*e^2*(d + e*x)^(5/2) - 270270*b^3*c*d^2*e^3*(d + e*x)^(5/2) - 810810*a*b*c^2*
d^2*e^3*(d + e*x)^(5/2) + 27027*b^4*d*e^4*(d + e*x)^(5/2) + 324324*a*b^2*c*d*e^4*(d + e*x)^(5/2) + 162162*a^2*
c^2*d*e^4*(d + e*x)^(5/2) - 27027*a*b^3*e^5*(d + e*x)^(5/2) - 81081*a^2*b*c*e^5*(d + e*x)^(5/2) - 450450*c^4*d
^4*(d + e*x)^(7/2) + 900900*b*c^3*d^3*e*(d + e*x)^(7/2) - 579150*b^2*c^2*d^2*e^2*(d + e*x)^(7/2) - 386100*a*c^
3*d^2*e^2*(d + e*x)^(7/2) + 128700*b^3*c*d*e^3*(d + e*x)^(7/2) + 386100*a*b*c^2*d*e^3*(d + e*x)^(7/2) - 6435*b
^4*e^4*(d + e*x)^(7/2) - 77220*a*b^2*c*e^4*(d + e*x)^(7/2) - 38610*a^2*c^2*e^4*(d + e*x)^(7/2) + 350350*c^4*d^
3*(d + e*x)^(9/2) - 525525*b*c^3*d^2*e*(d + e*x)^(9/2) + 225225*b^2*c^2*d*e^2*(d + e*x)^(9/2) + 150150*a*c^3*d
*e^2*(d + e*x)^(9/2) - 25025*b^3*c*e^3*(d + e*x)^(9/2) - 75075*a*b*c^2*e^3*(d + e*x)^(9/2) - 171990*c^4*d^2*(d
+ e*x)^(11/2) + 171990*b*c^3*d*e*(d + e*x)^(11/2) - 36855*b^2*c^2*e^2*(d + e*x)^(11/2) - 24570*a*c^3*e^2*(d +
e*x)^(11/2) + 48510*c^4*d*(d + e*x)^(13/2) - 24255*b*c^3*e*(d + e*x)^(13/2) - 6006*c^4*(d + e*x)^(15/2)))/(45
045*e^8)

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fricas [A]  time = 0.41, size = 648, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (6006 \, c^{4} e^{7} x^{7} - 28672 \, c^{4} d^{7} + 107520 \, b c^{3} d^{6} e + 45045 \, a^{3} b e^{7} - 49920 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 91520 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 20592 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 72072 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 30030 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 1617 \, {\left (4 \, c^{4} d e^{6} - 15 \, b c^{3} e^{7}\right )} x^{6} + 63 \, {\left (112 \, c^{4} d^{2} e^{5} - 420 \, b c^{3} d e^{6} + 195 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 35 \, {\left (224 \, c^{4} d^{3} e^{4} - 840 \, b c^{3} d^{2} e^{5} + 390 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 715 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + 5 \, {\left (1792 \, c^{4} d^{4} e^{3} - 6720 \, b c^{3} d^{3} e^{4} + 3120 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 5720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 1287 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{5} e^{2} - 13440 \, b c^{3} d^{4} e^{3} + 6240 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 11440 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 2574 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 9009 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + {\left (14336 \, c^{4} d^{6} e - 53760 \, b c^{3} d^{5} e^{2} + 24960 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 45760 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 10296 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 36036 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 15015 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\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((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(6006*c^4*e^7*x^7 - 28672*c^4*d^7 + 107520*b*c^3*d^6*e + 45045*a^3*b*e^7 - 49920*(3*b^2*c^2 + 2*a*c^3)
*d^5*e^2 + 91520*(b^3*c + 3*a*b*c^2)*d^4*e^3 - 20592*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 + 72072*(a*b^3 + 3
*a^2*b*c)*d^2*e^5 - 30030*(3*a^2*b^2 + 2*a^3*c)*d*e^6 - 1617*(4*c^4*d*e^6 - 15*b*c^3*e^7)*x^6 + 63*(112*c^4*d^
2*e^5 - 420*b*c^3*d*e^6 + 195*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 35*(224*c^4*d^3*e^4 - 840*b*c^3*d^2*e^5 + 390*(
3*b^2*c^2 + 2*a*c^3)*d*e^6 - 715*(b^3*c + 3*a*b*c^2)*e^7)*x^4 + 5*(1792*c^4*d^4*e^3 - 6720*b*c^3*d^3*e^4 + 312
0*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 5720*(b^3*c + 3*a*b*c^2)*d*e^6 + 1287*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^
3 - 3*(3584*c^4*d^5*e^2 - 13440*b*c^3*d^4*e^3 + 6240*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 11440*(b^3*c + 3*a*b*c^2)
*d^2*e^5 + 2574*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 9009*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 + (14336*c^4*d^6*e -
53760*b*c^3*d^5*e^2 + 24960*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 45760*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 10296*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 36036*(a*b^3 + 3*a^2*b*c)*d*e^6 + 15015*(3*a^2*b^2 + 2*a^3*c)*e^7)*x)*sqrt(e*
x + d)/e^8

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giac [B]  time = 0.23, size = 841, normalized size = 1.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b^2*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)
*d)*a^3*c*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^3*e^(-2) + 27027
*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b*c*e^(-2) + 1287*(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^4*e^(-3) + 15444*(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*b^2*c*e^(-3) + 7722*(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^2*c^2*e^(-3) + 715*(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^3*c
*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*b*c^2*e^(-4) + 585*(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^2*c^2*e^(-5) + 390*(6
3*(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*c^3*e^(-5) + 105*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 500
5*(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*c^3*e^(-6) + 14*(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)*c^4*e^(-7) + 45045*sqrt(x*e + d)*a^3*b)*e^(-1)

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maple [B]  time = 0.06, size = 795, normalized size = 1.87 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (6006 c^{4} e^{7} x^{7}+24255 b \,c^{3} e^{7} x^{6}-6468 c^{4} d \,e^{6} x^{6}+24570 a \,c^{3} e^{7} x^{5}+36855 b^{2} c^{2} e^{7} x^{5}-26460 b \,c^{3} d \,e^{6} x^{5}+7056 c^{4} d^{2} e^{5} x^{5}+75075 a b \,c^{2} e^{7} x^{4}-27300 a \,c^{3} d \,e^{6} x^{4}+25025 b^{3} c \,e^{7} x^{4}-40950 b^{2} c^{2} d \,e^{6} x^{4}+29400 b \,c^{3} d^{2} e^{5} x^{4}-7840 c^{4} d^{3} e^{4} x^{4}+38610 a^{2} c^{2} e^{7} x^{3}+77220 a \,b^{2} c \,e^{7} x^{3}-85800 a b \,c^{2} d \,e^{6} x^{3}+31200 a \,c^{3} d^{2} e^{5} x^{3}+6435 b^{4} e^{7} x^{3}-28600 b^{3} c d \,e^{6} x^{3}+46800 b^{2} c^{2} d^{2} e^{5} x^{3}-33600 b \,c^{3} d^{3} e^{4} x^{3}+8960 c^{4} d^{4} e^{3} x^{3}+81081 a^{2} b c \,e^{7} x^{2}-46332 a^{2} c^{2} d \,e^{6} x^{2}+27027 a \,b^{3} e^{7} x^{2}-92664 a \,b^{2} c d \,e^{6} x^{2}+102960 a b \,c^{2} d^{2} e^{5} x^{2}-37440 a \,c^{3} d^{3} e^{4} x^{2}-7722 b^{4} d \,e^{6} x^{2}+34320 b^{3} c \,d^{2} e^{5} x^{2}-56160 b^{2} c^{2} d^{3} e^{4} x^{2}+40320 b \,c^{3} d^{4} e^{3} x^{2}-10752 c^{4} d^{5} e^{2} x^{2}+30030 a^{3} c \,e^{7} x +45045 a^{2} b^{2} e^{7} x -108108 a^{2} b c d \,e^{6} x +61776 a^{2} c^{2} d^{2} e^{5} x -36036 a \,b^{3} d \,e^{6} x +123552 a \,b^{2} c \,d^{2} e^{5} x -137280 a b \,c^{2} d^{3} e^{4} x +49920 a \,c^{3} d^{4} e^{3} x +10296 b^{4} d^{2} e^{5} x -45760 b^{3} c \,d^{3} e^{4} x +74880 b^{2} c^{2} d^{4} e^{3} x -53760 b \,c^{3} d^{5} e^{2} x +14336 c^{4} d^{6} e x +45045 b \,a^{3} e^{7}-60060 a^{3} c d \,e^{6}-90090 a^{2} b^{2} d \,e^{6}+216216 a^{2} b c \,d^{2} e^{5}-123552 a^{2} c^{2} d^{3} e^{4}+72072 a \,b^{3} d^{2} e^{5}-247104 a \,b^{2} c \,d^{3} e^{4}+274560 a b \,c^{2} d^{4} e^{3}-99840 a \,c^{3} d^{5} e^{2}-20592 b^{4} d^{3} e^{4}+91520 b^{3} c \,d^{4} e^{3}-149760 b^{2} c^{2} d^{5} e^{2}+107520 b \,c^{3} d^{6} e -28672 c^{4} d^{7}\right )}{45045 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(1/2)*(6006*c^4*e^7*x^7+24255*b*c^3*e^7*x^6-6468*c^4*d*e^6*x^6+24570*a*c^3*e^7*x^5+36855*b^2*c
^2*e^7*x^5-26460*b*c^3*d*e^6*x^5+7056*c^4*d^2*e^5*x^5+75075*a*b*c^2*e^7*x^4-27300*a*c^3*d*e^6*x^4+25025*b^3*c*
e^7*x^4-40950*b^2*c^2*d*e^6*x^4+29400*b*c^3*d^2*e^5*x^4-7840*c^4*d^3*e^4*x^4+38610*a^2*c^2*e^7*x^3+77220*a*b^2
*c*e^7*x^3-85800*a*b*c^2*d*e^6*x^3+31200*a*c^3*d^2*e^5*x^3+6435*b^4*e^7*x^3-28600*b^3*c*d*e^6*x^3+46800*b^2*c^
2*d^2*e^5*x^3-33600*b*c^3*d^3*e^4*x^3+8960*c^4*d^4*e^3*x^3+81081*a^2*b*c*e^7*x^2-46332*a^2*c^2*d*e^6*x^2+27027
*a*b^3*e^7*x^2-92664*a*b^2*c*d*e^6*x^2+102960*a*b*c^2*d^2*e^5*x^2-37440*a*c^3*d^3*e^4*x^2-7722*b^4*d*e^6*x^2+3
4320*b^3*c*d^2*e^5*x^2-56160*b^2*c^2*d^3*e^4*x^2+40320*b*c^3*d^4*e^3*x^2-10752*c^4*d^5*e^2*x^2+30030*a^3*c*e^7
*x+45045*a^2*b^2*e^7*x-108108*a^2*b*c*d*e^6*x+61776*a^2*c^2*d^2*e^5*x-36036*a*b^3*d*e^6*x+123552*a*b^2*c*d^2*e
^5*x-137280*a*b*c^2*d^3*e^4*x+49920*a*c^3*d^4*e^3*x+10296*b^4*d^2*e^5*x-45760*b^3*c*d^3*e^4*x+74880*b^2*c^2*d^
4*e^3*x-53760*b*c^3*d^5*e^2*x+14336*c^4*d^6*e*x+45045*a^3*b*e^7-60060*a^3*c*d*e^6-90090*a^2*b^2*d*e^6+216216*a
^2*b*c*d^2*e^5-123552*a^2*c^2*d^3*e^4+72072*a*b^3*d^2*e^5-247104*a*b^2*c*d^3*e^4+274560*a*b*c^2*d^4*e^3-99840*
a*c^3*d^5*e^2-20592*b^4*d^3*e^4+91520*b^3*c*d^4*e^3-149760*b^2*c^2*d^5*e^2+107520*b*c^3*d^6*e-28672*c^4*d^7)/e
^8

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maxima [A]  time = 0.55, size = 645, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (6006 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{4} - 24255 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 25025 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 45045 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\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((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(6006*(e*x + d)^(15/2)*c^4 - 24255*(2*c^4*d - b*c^3*e)*(e*x + d)^(13/2) + 12285*(14*c^4*d^2 - 14*b*c^3
*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(11/2) - 25025*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c
^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(9/2) + 6435*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*
a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(7/2) - 27027*(1
4*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(5/2) + 15015*(14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(
3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 6*(
a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(3/2) - 45045*(2*c^4*d^7 - 7*b*c^3*d^6*e - a^3
*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*
e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*sqrt(e*x + d))/e^8

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mupad [B]  time = 1.93, size = 444, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{11\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{7\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{5\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{9\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(11/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(11*e^8) + (4*c^4*(d + e*x)^(15
/2))/(15*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(13/2))/(13*e^8) + ((d + e*x)^(7/2)*(2*b^4*e^4 + 140*c^4*d^
4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*e
^3 - 120*a*b*c^2*d*e^3))/(7*e^8) + (2*(b*e - 2*c*d)*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e)^3)/e^8 + (6*(b*e -
2*c*d)*(d + e*x)^(5/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d^2*e^2
- 14*b*c^2*d^3*e - 10*a*b*c*d*e^3))/(5*e^8) + (2*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^2*(3*b^2*e^2 + 14*c^2
*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(3*e^8) + (10*c*(b*e - 2*c*d)*(d + e*x)^(9/2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2
- 7*b*c*d*e))/(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((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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