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

Optimal. Leaf size=282 \[ \frac {6 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac {6 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 \sqrt {d+e x}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}-\frac {6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {698} \begin {gather*} \frac {6 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac {6 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 \sqrt {d+e x}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}-\frac {6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7*Sqrt
[d + e*x]) + (6*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*Sqrt[d + e*x])/e^7 - (2*(2*c
*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(3/2))/(3*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2
- c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*
x)^(9/2))/(9*e^7)

Rule 698

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

Rubi steps

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

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Mathematica [A]  time = 0.37, size = 395, normalized size = 1.40 \begin {gather*} \frac {2 \left (63 c e^2 \left (5 a^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 a b e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-105 e^3 \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )-9 c^2 e \left (5 b \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )-7 a e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+5 c^3 \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )}{315 e^7 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*
x^6) - 105*e^3*(a^3*e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b^3*(16*d^3 +
24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 63*c*e^2*(5*a^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*a*b*e*(-16*d^3
- 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4
)) - 9*c^2*e*(-7*a*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 5*b*(256*d^5 + 384*d
^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5))))/(315*e^7*(d + e*x)^(3/2))

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IntegrateAlgebraic [B]  time = 0.25, size = 592, normalized size = 2.10 \begin {gather*} \frac {2 \left (-105 a^3 e^6-945 a^2 b e^5 (d+e x)+315 a^2 b d e^5-315 a^2 c d^2 e^4+1890 a^2 c d e^4 (d+e x)+945 a^2 c e^4 (d+e x)^2-315 a b^2 d^2 e^4+1890 a b^2 d e^4 (d+e x)+945 a b^2 e^4 (d+e x)^2+630 a b c d^3 e^3-5670 a b c d^2 e^3 (d+e x)-5670 a b c d e^3 (d+e x)^2+630 a b c e^3 (d+e x)^3-315 a c^2 d^4 e^2+3780 a c^2 d^3 e^2 (d+e x)+5670 a c^2 d^2 e^2 (d+e x)^2-1260 a c^2 d e^2 (d+e x)^3+189 a c^2 e^2 (d+e x)^4+105 b^3 d^3 e^3-945 b^3 d^2 e^3 (d+e x)-945 b^3 d e^3 (d+e x)^2+105 b^3 e^3 (d+e x)^3-315 b^2 c d^4 e^2+3780 b^2 c d^3 e^2 (d+e x)+5670 b^2 c d^2 e^2 (d+e x)^2-1260 b^2 c d e^2 (d+e x)^3+189 b^2 c e^2 (d+e x)^4+315 b c^2 d^5 e-4725 b c^2 d^4 e (d+e x)-9450 b c^2 d^3 e (d+e x)^2+3150 b c^2 d^2 e (d+e x)^3-945 b c^2 d e (d+e x)^4+135 b c^2 e (d+e x)^5-105 c^3 d^6+1890 c^3 d^5 (d+e x)+4725 c^3 d^4 (d+e x)^2-2100 c^3 d^3 (d+e x)^3+945 c^3 d^2 (d+e x)^4-270 c^3 d (d+e x)^5+35 c^3 (d+e x)^6\right )}{315 e^7 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-105*c^3*d^6 + 315*b*c^2*d^5*e - 315*b^2*c*d^4*e^2 - 315*a*c^2*d^4*e^2 + 105*b^3*d^3*e^3 + 630*a*b*c*d^3*e
^3 - 315*a*b^2*d^2*e^4 - 315*a^2*c*d^2*e^4 + 315*a^2*b*d*e^5 - 105*a^3*e^6 + 1890*c^3*d^5*(d + e*x) - 4725*b*c
^2*d^4*e*(d + e*x) + 3780*b^2*c*d^3*e^2*(d + e*x) + 3780*a*c^2*d^3*e^2*(d + e*x) - 945*b^3*d^2*e^3*(d + e*x) -
5670*a*b*c*d^2*e^3*(d + e*x) + 1890*a*b^2*d*e^4*(d + e*x) + 1890*a^2*c*d*e^4*(d + e*x) - 945*a^2*b*e^5*(d + e
*x) + 4725*c^3*d^4*(d + e*x)^2 - 9450*b*c^2*d^3*e*(d + e*x)^2 + 5670*b^2*c*d^2*e^2*(d + e*x)^2 + 5670*a*c^2*d^
2*e^2*(d + e*x)^2 - 945*b^3*d*e^3*(d + e*x)^2 - 5670*a*b*c*d*e^3*(d + e*x)^2 + 945*a*b^2*e^4*(d + e*x)^2 + 945
*a^2*c*e^4*(d + e*x)^2 - 2100*c^3*d^3*(d + e*x)^3 + 3150*b*c^2*d^2*e*(d + e*x)^3 - 1260*b^2*c*d*e^2*(d + e*x)^
3 - 1260*a*c^2*d*e^2*(d + e*x)^3 + 105*b^3*e^3*(d + e*x)^3 + 630*a*b*c*e^3*(d + e*x)^3 + 945*c^3*d^2*(d + e*x)
^4 - 945*b*c^2*d*e*(d + e*x)^4 + 189*b^2*c*e^2*(d + e*x)^4 + 189*a*c^2*e^2*(d + e*x)^4 - 270*c^3*d*(d + e*x)^5
+ 135*b*c^2*e*(d + e*x)^5 + 35*c^3*(d + e*x)^6))/(315*e^7*(d + e*x)^(3/2))

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fricas [A]  time = 0.41, size = 429, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (35 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 11520 \, b c^{2} d^{5} e - 630 \, a^{2} b d e^{5} - 105 \, a^{3} e^{6} + 8064 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 1680 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2520 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 15 \, {\left (4 \, c^{3} d e^{5} - 9 \, b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (40 \, c^{3} d^{2} e^{4} - 90 \, b c^{2} d e^{5} + 63 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - {\left (320 \, c^{3} d^{3} e^{3} - 720 \, b c^{2} d^{2} e^{4} + 504 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 105 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} - 1440 \, b c^{2} d^{3} e^{3} + 1008 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 210 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 315 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 3 \, {\left (2560 \, c^{3} d^{5} e - 5760 \, b c^{2} d^{4} e^{2} - 315 \, a^{2} b e^{6} + 4032 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 840 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 1260 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*c^3*e^6*x^6 + 5120*c^3*d^6 - 11520*b*c^2*d^5*e - 630*a^2*b*d*e^5 - 105*a^3*e^6 + 8064*(b^2*c + a*c^2
)*d^4*e^2 - 1680*(b^3 + 6*a*b*c)*d^3*e^3 + 2520*(a*b^2 + a^2*c)*d^2*e^4 - 15*(4*c^3*d*e^5 - 9*b*c^2*e^6)*x^5 +
3*(40*c^3*d^2*e^4 - 90*b*c^2*d*e^5 + 63*(b^2*c + a*c^2)*e^6)*x^4 - (320*c^3*d^3*e^3 - 720*b*c^2*d^2*e^4 + 504
*(b^2*c + a*c^2)*d*e^5 - 105*(b^3 + 6*a*b*c)*e^6)*x^3 + 3*(640*c^3*d^4*e^2 - 1440*b*c^2*d^3*e^3 + 1008*(b^2*c
+ a*c^2)*d^2*e^4 - 210*(b^3 + 6*a*b*c)*d*e^5 + 315*(a*b^2 + a^2*c)*e^6)*x^2 + 3*(2560*c^3*d^5*e - 5760*b*c^2*d
^4*e^2 - 315*a^2*b*e^6 + 4032*(b^2*c + a*c^2)*d^3*e^3 - 840*(b^3 + 6*a*b*c)*d^2*e^4 + 1260*(a*b^2 + a^2*c)*d*e
^5)*x)*sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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giac [B]  time = 0.24, size = 612, normalized size = 2.17 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} e^{56} - 270 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e^{56} + 945 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e^{56} - 2100 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt {x e + d} c^{3} d^{4} e^{56} + 135 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{57} - 945 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{57} + 3150 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{57} - 9450 \, \sqrt {x e + d} b c^{2} d^{3} e^{57} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{58} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} e^{58} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{58} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d e^{58} + 5670 \, \sqrt {x e + d} b^{2} c d^{2} e^{58} + 5670 \, \sqrt {x e + d} a c^{2} d^{2} e^{58} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{59} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c e^{59} - 945 \, \sqrt {x e + d} b^{3} d e^{59} - 5670 \, \sqrt {x e + d} a b c d e^{59} + 945 \, \sqrt {x e + d} a b^{2} e^{60} + 945 \, \sqrt {x e + d} a^{2} c e^{60}\right )} e^{\left (-63\right )} + \frac {2 \, {\left (18 \, {\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 45 \, {\left (x e + d\right )} b c^{2} d^{4} e + 3 \, b c^{2} d^{5} e + 36 \, {\left (x e + d\right )} b^{2} c d^{3} e^{2} + 36 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{4} e^{2} - 3 \, a c^{2} d^{4} e^{2} - 9 \, {\left (x e + d\right )} b^{3} d^{2} e^{3} - 54 \, {\left (x e + d\right )} a b c d^{2} e^{3} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 18 \, {\left (x e + d\right )} a b^{2} d e^{4} + 18 \, {\left (x e + d\right )} a^{2} c d e^{4} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 9 \, {\left (x e + d\right )} a^{2} b e^{5} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*e^56 - 270*(x*e + d)^(7/2)*c^3*d*e^56 + 945*(x*e + d)^(5/2)*c^3*d^2*e^56 - 2100*
(x*e + d)^(3/2)*c^3*d^3*e^56 + 4725*sqrt(x*e + d)*c^3*d^4*e^56 + 135*(x*e + d)^(7/2)*b*c^2*e^57 - 945*(x*e + d
)^(5/2)*b*c^2*d*e^57 + 3150*(x*e + d)^(3/2)*b*c^2*d^2*e^57 - 9450*sqrt(x*e + d)*b*c^2*d^3*e^57 + 189*(x*e + d)
^(5/2)*b^2*c*e^58 + 189*(x*e + d)^(5/2)*a*c^2*e^58 - 1260*(x*e + d)^(3/2)*b^2*c*d*e^58 - 1260*(x*e + d)^(3/2)*
a*c^2*d*e^58 + 5670*sqrt(x*e + d)*b^2*c*d^2*e^58 + 5670*sqrt(x*e + d)*a*c^2*d^2*e^58 + 105*(x*e + d)^(3/2)*b^3
*e^59 + 630*(x*e + d)^(3/2)*a*b*c*e^59 - 945*sqrt(x*e + d)*b^3*d*e^59 - 5670*sqrt(x*e + d)*a*b*c*d*e^59 + 945*
sqrt(x*e + d)*a*b^2*e^60 + 945*sqrt(x*e + d)*a^2*c*e^60)*e^(-63) + 2/3*(18*(x*e + d)*c^3*d^5 - c^3*d^6 - 45*(x
*e + d)*b*c^2*d^4*e + 3*b*c^2*d^5*e + 36*(x*e + d)*b^2*c*d^3*e^2 + 36*(x*e + d)*a*c^2*d^3*e^2 - 3*b^2*c*d^4*e^
2 - 3*a*c^2*d^4*e^2 - 9*(x*e + d)*b^3*d^2*e^3 - 54*(x*e + d)*a*b*c*d^2*e^3 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 1
8*(x*e + d)*a*b^2*d*e^4 + 18*(x*e + d)*a^2*c*d*e^4 - 3*a*b^2*d^2*e^4 - 3*a^2*c*d^2*e^4 - 9*(x*e + d)*a^2*b*e^5
+ 3*a^2*b*d*e^5 - a^3*e^6)*e^(-7)/(x*e + d)^(3/2)

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maple [A]  time = 0.05, size = 495, normalized size = 1.76 \begin {gather*} -\frac {2 \left (-35 c^{3} x^{6} e^{6}-135 b \,c^{2} e^{6} x^{5}+60 c^{3} d \,e^{5} x^{5}-189 a \,c^{2} e^{6} x^{4}-189 b^{2} c \,e^{6} x^{4}+270 b \,c^{2} d \,e^{5} x^{4}-120 c^{3} d^{2} e^{4} x^{4}-630 a b c \,e^{6} x^{3}+504 a \,c^{2} d \,e^{5} x^{3}-105 b^{3} e^{6} x^{3}+504 b^{2} c d \,e^{5} x^{3}-720 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}-945 a^{2} c \,e^{6} x^{2}-945 a \,b^{2} e^{6} x^{2}+3780 a b c d \,e^{5} x^{2}-3024 a \,c^{2} d^{2} e^{4} x^{2}+630 b^{3} d \,e^{5} x^{2}-3024 b^{2} c \,d^{2} e^{4} x^{2}+4320 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}+945 a^{2} b \,e^{6} x -3780 a^{2} c d \,e^{5} x -3780 a \,b^{2} d \,e^{5} x +15120 a b c \,d^{2} e^{4} x -12096 a \,c^{2} d^{3} e^{3} x +2520 b^{3} d^{2} e^{4} x -12096 b^{2} c \,d^{3} e^{3} x +17280 b \,c^{2} d^{4} e^{2} x -7680 c^{3} d^{5} e x +105 a^{3} e^{6}+630 a^{2} b d \,e^{5}-2520 a^{2} c \,d^{2} e^{4}-2520 a \,b^{2} d^{2} e^{4}+10080 a b c \,d^{3} e^{3}-8064 a \,c^{2} d^{4} e^{2}+1680 b^{3} d^{3} e^{3}-8064 b^{2} c \,d^{4} e^{2}+11520 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x)

[Out]

-2/315/(e*x+d)^(3/2)*(-35*c^3*e^6*x^6-135*b*c^2*e^6*x^5+60*c^3*d*e^5*x^5-189*a*c^2*e^6*x^4-189*b^2*c*e^6*x^4+2
70*b*c^2*d*e^5*x^4-120*c^3*d^2*e^4*x^4-630*a*b*c*e^6*x^3+504*a*c^2*d*e^5*x^3-105*b^3*e^6*x^3+504*b^2*c*d*e^5*x
^3-720*b*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3-945*a^2*c*e^6*x^2-945*a*b^2*e^6*x^2+3780*a*b*c*d*e^5*x^2-3024*a*c
^2*d^2*e^4*x^2+630*b^3*d*e^5*x^2-3024*b^2*c*d^2*e^4*x^2+4320*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2+945*a^2*b*
e^6*x-3780*a^2*c*d*e^5*x-3780*a*b^2*d*e^5*x+15120*a*b*c*d^2*e^4*x-12096*a*c^2*d^3*e^3*x+2520*b^3*d^2*e^4*x-120
96*b^2*c*d^3*e^3*x+17280*b*c^2*d^4*e^2*x-7680*c^3*d^5*e*x+105*a^3*e^6+630*a^2*b*d*e^5-2520*a^2*c*d^2*e^4-2520*
a*b^2*d^2*e^4+10080*a*b*c*d^3*e^3-8064*a*c^2*d^4*e^2+1680*b^3*d^3*e^3-8064*b^2*c*d^4*e^2+11520*b*c^2*d^5*e-512
0*c^3*d^6)/e^7

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maxima [A]  time = 0.97, size = 413, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} - 135 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 945 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {105 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 9 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{6}}\right )}}{315 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((35*(e*x + d)^(9/2)*c^3 - 135*(2*c^3*d - b*c^2*e)*(e*x + d)^(7/2) + 189*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2
*c + a*c^2)*e^2)*(e*x + d)^(5/2) - 105*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*
c)*e^3)*(e*x + d)^(3/2) + 945*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3
+ (a*b^2 + a^2*c)*e^4)*sqrt(e*x + d))/e^6 - 105*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c
+ a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4 - 9*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*
e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d))/((e*x + d)^(3/
2)*e^6))/e

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mupad [B]  time = 0.08, size = 448, normalized size = 1.59 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\frac {2\,a^3\,e^6}{3}-\left (d+e\,x\right )\,\left (-6\,a^2\,b\,e^5+12\,a^2\,c\,d\,e^4+12\,a\,b^2\,d\,e^4-36\,a\,b\,c\,d^2\,e^3+24\,a\,c^2\,d^3\,e^2-6\,b^3\,d^2\,e^3+24\,b^2\,c\,d^3\,e^2-30\,b\,c^2\,d^4\,e+12\,c^3\,d^5\right )+\frac {2\,c^3\,d^6}{3}-\frac {2\,b^3\,d^3\,e^3}{3}+2\,a\,b^2\,d^2\,e^4+2\,a\,c^2\,d^4\,e^2+2\,a^2\,c\,d^2\,e^4+2\,b^2\,c\,d^4\,e^2-2\,a^2\,b\,d\,e^5-2\,b\,c^2\,d^5\,e-4\,a\,b\,c\,d^3\,e^3}{e^7\,{\left (d+e\,x\right )}^{3/2}}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{5\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{3\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^(5/2),x)

[Out]

((d + e*x)^(1/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 -
60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/e^7 + (2*c^3*(d + e*x)^(9/2))/(9*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(
7/2))/(7*e^7) - ((2*a^3*e^6)/3 - (d + e*x)*(12*c^3*d^5 - 6*a^2*b*e^5 - 6*b^3*d^2*e^3 + 24*a*c^2*d^3*e^2 + 24*b
^2*c*d^3*e^2 + 12*a*b^2*d*e^4 + 12*a^2*c*d*e^4 - 30*b*c^2*d^4*e - 36*a*b*c*d^2*e^3) + (2*c^3*d^6)/3 - (2*b^3*d
^3*e^3)/3 + 2*a*b^2*d^2*e^4 + 2*a*c^2*d^4*e^2 + 2*a^2*c*d^2*e^4 + 2*b^2*c*d^4*e^2 - 2*a^2*b*d*e^5 - 2*b*c^2*d^
5*e - 4*a*b*c*d^3*e^3)/(e^7*(d + e*x)^(3/2)) + ((d + e*x)^(5/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b
*c^2*d*e))/(5*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(3*e^7)

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sympy [A]  time = 162.14, size = 348, normalized size = 1.23 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (6 a c^{2} e^{2} + 6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (6 a^{2} c e^{4} + 6 a b^{2} e^{4} - 36 a b c d e^{3} + 36 a c^{2} d^{2} e^{2} - 6 b^{3} d e^{3} + 36 b^{2} c d^{2} e^{2} - 60 b c^{2} d^{3} e + 30 c^{3} d^{4}\right )}{e^{7}} - \frac {6 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \sqrt {d + e x}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**(5/2),x)

[Out]

2*c**3*(d + e*x)**(9/2)/(9*e**7) + (d + e*x)**(7/2)*(6*b*c**2*e - 12*c**3*d)/(7*e**7) + (d + e*x)**(5/2)*(6*a*
c**2*e**2 + 6*b**2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(5*e**7) + (d + e*x)**(3/2)*(12*a*b*c*e**3 - 24*a*c*
*2*d*e**2 + 2*b**3*e**3 - 24*b**2*c*d*e**2 + 60*b*c**2*d**2*e - 40*c**3*d**3)/(3*e**7) + sqrt(d + e*x)*(6*a**2
*c*e**4 + 6*a*b**2*e**4 - 36*a*b*c*d*e**3 + 36*a*c**2*d**2*e**2 - 6*b**3*d*e**3 + 36*b**2*c*d**2*e**2 - 60*b*c
**2*d**3*e + 30*c**3*d**4)/e**7 - 6*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(e**7*sqrt(d + e*x)) - 2*(a*e**
2 - b*d*e + c*d**2)**3/(3*e**7*(d + e*x)**(3/2))

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