3.20.57 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=278 \[ \frac {2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac {6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 278, 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 {2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (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 )}{e^7}-\frac {6 \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 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

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

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Mathematica [A]  time = 0.37, size = 391, normalized size = 1.41 \begin {gather*} -\frac {2 \left (7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )+7 e^3 \left (a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )-\left (b^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )-7 c^2 e \left (a e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+b \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6
*x^6) + 7*e^3*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - b^3*(16*d^3 + 40*
d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + 7*c*e^2*(a^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*b*e*(16*d^3 + 40
*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + b^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4
)) - 7*c^2*e*(a*e*(-128*d^4 - 320*d^3*e*x - 240*d^2*e^2*x^2 - 40*d*e^3*x^3 + 5*e^4*x^4) + b*(256*d^5 + 640*d^4
*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(35*e^7*(d + e*x)^(5/2))

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IntegrateAlgebraic [B]  time = 0.23, size = 592, normalized size = 2.13 \begin {gather*} \frac {2 \left (-7 a^3 e^6-35 a^2 b e^5 (d+e x)+21 a^2 b d e^5-21 a^2 c d^2 e^4+70 a^2 c d e^4 (d+e x)-105 a^2 c e^4 (d+e x)^2-21 a b^2 d^2 e^4+70 a b^2 d e^4 (d+e x)-105 a b^2 e^4 (d+e x)^2+42 a b c d^3 e^3-210 a b c d^2 e^3 (d+e x)+630 a b c d e^3 (d+e x)^2+210 a b c e^3 (d+e x)^3-21 a c^2 d^4 e^2+140 a c^2 d^3 e^2 (d+e x)-630 a c^2 d^2 e^2 (d+e x)^2-420 a c^2 d e^2 (d+e x)^3+35 a c^2 e^2 (d+e x)^4+7 b^3 d^3 e^3-35 b^3 d^2 e^3 (d+e x)+105 b^3 d e^3 (d+e x)^2+35 b^3 e^3 (d+e x)^3-21 b^2 c d^4 e^2+140 b^2 c d^3 e^2 (d+e x)-630 b^2 c d^2 e^2 (d+e x)^2-420 b^2 c d e^2 (d+e x)^3+35 b^2 c e^2 (d+e x)^4+21 b c^2 d^5 e-175 b c^2 d^4 e (d+e x)+1050 b c^2 d^3 e (d+e x)^2+1050 b c^2 d^2 e (d+e x)^3-175 b c^2 d e (d+e x)^4+21 b c^2 e (d+e x)^5-7 c^3 d^6+70 c^3 d^5 (d+e x)-525 c^3 d^4 (d+e x)^2-700 c^3 d^3 (d+e x)^3+175 c^3 d^2 (d+e x)^4-42 c^3 d (d+e x)^5+5 c^3 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(2*(-7*c^3*d^6 + 21*b*c^2*d^5*e - 21*b^2*c*d^4*e^2 - 21*a*c^2*d^4*e^2 + 7*b^3*d^3*e^3 + 42*a*b*c*d^3*e^3 - 21*
a*b^2*d^2*e^4 - 21*a^2*c*d^2*e^4 + 21*a^2*b*d*e^5 - 7*a^3*e^6 + 70*c^3*d^5*(d + e*x) - 175*b*c^2*d^4*e*(d + e*
x) + 140*b^2*c*d^3*e^2*(d + e*x) + 140*a*c^2*d^3*e^2*(d + e*x) - 35*b^3*d^2*e^3*(d + e*x) - 210*a*b*c*d^2*e^3*
(d + e*x) + 70*a*b^2*d*e^4*(d + e*x) + 70*a^2*c*d*e^4*(d + e*x) - 35*a^2*b*e^5*(d + e*x) - 525*c^3*d^4*(d + e*
x)^2 + 1050*b*c^2*d^3*e*(d + e*x)^2 - 630*b^2*c*d^2*e^2*(d + e*x)^2 - 630*a*c^2*d^2*e^2*(d + e*x)^2 + 105*b^3*
d*e^3*(d + e*x)^2 + 630*a*b*c*d*e^3*(d + e*x)^2 - 105*a*b^2*e^4*(d + e*x)^2 - 105*a^2*c*e^4*(d + e*x)^2 - 700*
c^3*d^3*(d + e*x)^3 + 1050*b*c^2*d^2*e*(d + e*x)^3 - 420*b^2*c*d*e^2*(d + e*x)^3 - 420*a*c^2*d*e^2*(d + e*x)^3
+ 35*b^3*e^3*(d + e*x)^3 + 210*a*b*c*e^3*(d + e*x)^3 + 175*c^3*d^2*(d + e*x)^4 - 175*b*c^2*d*e*(d + e*x)^4 +
35*b^2*c*e^2*(d + e*x)^4 + 35*a*c^2*e^2*(d + e*x)^4 - 42*c^3*d*(d + e*x)^5 + 21*b*c^2*e*(d + e*x)^5 + 5*c^3*(d
+ e*x)^6))/(35*e^7*(d + e*x)^(5/2))

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fricas [A]  time = 0.43, size = 440, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 14 \, a^{2} b d e^{5} - 7 \, a^{3} e^{6} - 896 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 112 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 56 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \, {\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 7 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 15 \, {\left (128 \, c^{3} d^{4} e^{2} - 224 \, b c^{2} d^{3} e^{3} + 112 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 14 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 7 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \, {\left (512 \, c^{3} d^{5} e - 896 \, b c^{2} d^{4} e^{2} + 7 \, a^{2} b e^{6} + 448 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 56 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 28 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*e^6*x^6 - 1024*c^3*d^6 + 1792*b*c^2*d^5*e - 14*a^2*b*d*e^5 - 7*a^3*e^6 - 896*(b^2*c + a*c^2)*d^4*e
^2 + 112*(b^3 + 6*a*b*c)*d^3*e^3 - 56*(a*b^2 + a^2*c)*d^2*e^4 - 3*(4*c^3*d*e^5 - 7*b*c^2*e^6)*x^5 + 5*(8*c^3*d
^2*e^4 - 14*b*c^2*d*e^5 + 7*(b^2*c + a*c^2)*e^6)*x^4 - 5*(64*c^3*d^3*e^3 - 112*b*c^2*d^2*e^4 + 56*(b^2*c + a*c
^2)*d*e^5 - 7*(b^3 + 6*a*b*c)*e^6)*x^3 - 15*(128*c^3*d^4*e^2 - 224*b*c^2*d^3*e^3 + 112*(b^2*c + a*c^2)*d^2*e^4
- 14*(b^3 + 6*a*b*c)*d*e^5 + 7*(a*b^2 + a^2*c)*e^6)*x^2 - 5*(512*c^3*d^5*e - 896*b*c^2*d^4*e^2 + 7*a^2*b*e^6
+ 448*(b^2*c + a*c^2)*d^3*e^3 - 56*(b^3 + 6*a*b*c)*d^2*e^4 + 28*(a*b^2 + a^2*c)*d*e^5)*x)*sqrt(e*x + d)/(e^10*
x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [B]  time = 0.24, size = 609, normalized size = 2.19 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {x e + d} c^{3} d^{3} e^{42} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} e^{43} - 175 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt {x e + d} b c^{2} d^{2} e^{43} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c e^{44} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} e^{44} - 420 \, \sqrt {x e + d} b^{2} c d e^{44} - 420 \, \sqrt {x e + d} a c^{2} d e^{44} + 35 \, \sqrt {x e + d} b^{3} e^{45} + 210 \, \sqrt {x e + d} a b c e^{45}\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \, {\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \, {\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \, {\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} + 90 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \, {\left (x e + d\right )} b^{2} c d^{3} e^{2} - 20 \, {\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} - 15 \, {\left (x e + d\right )}^{2} b^{3} d e^{3} - 90 \, {\left (x e + d\right )}^{2} a b c d e^{3} + 5 \, {\left (x e + d\right )} b^{3} d^{2} e^{3} + 30 \, {\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} + 15 \, {\left (x e + d\right )}^{2} a b^{2} e^{4} + 15 \, {\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \, {\left (x e + d\right )} a b^{2} d e^{4} - 10 \, {\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} + 5 \, {\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 )}}{5 \, {\left (x e + d\right )}^{\frac {5}{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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*e^42 - 42*(x*e + d)^(5/2)*c^3*d*e^42 + 175*(x*e + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt
(x*e + d)*c^3*d^3*e^42 + 21*(x*e + d)^(5/2)*b*c^2*e^43 - 175*(x*e + d)^(3/2)*b*c^2*d*e^43 + 1050*sqrt(x*e + d)
*b*c^2*d^2*e^43 + 35*(x*e + d)^(3/2)*b^2*c*e^44 + 35*(x*e + d)^(3/2)*a*c^2*e^44 - 420*sqrt(x*e + d)*b^2*c*d*e^
44 - 420*sqrt(x*e + d)*a*c^2*d*e^44 + 35*sqrt(x*e + d)*b^3*e^45 + 210*sqrt(x*e + d)*a*b*c*e^45)*e^(-49) - 2/5*
(75*(x*e + d)^2*c^3*d^4 - 10*(x*e + d)*c^3*d^5 + c^3*d^6 - 150*(x*e + d)^2*b*c^2*d^3*e + 25*(x*e + d)*b*c^2*d^
4*e - 3*b*c^2*d^5*e + 90*(x*e + d)^2*b^2*c*d^2*e^2 + 90*(x*e + d)^2*a*c^2*d^2*e^2 - 20*(x*e + d)*b^2*c*d^3*e^2
- 20*(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 - 15*(x*e + d)^2*b^3*d*e^3 - 90*(x*e + d)^2*
a*b*c*d*e^3 + 5*(x*e + d)*b^3*d^2*e^3 + 30*(x*e + d)*a*b*c*d^2*e^3 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 15*(x*e +
d)^2*a*b^2*e^4 + 15*(x*e + d)^2*a^2*c*e^4 - 10*(x*e + d)*a*b^2*d*e^4 - 10*(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 + 5*(x*e + d)*a^2*b*e^5 - 3*a^2*b*d*e^5 + a^3*e^6)*e^(-7)/(x*e + d)^(5/2)

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maple [A]  time = 0.06, size = 495, normalized size = 1.78 \begin {gather*} -\frac {2 \left (-5 c^{3} x^{6} e^{6}-21 b \,c^{2} e^{6} x^{5}+12 c^{3} d \,e^{5} x^{5}-35 a \,c^{2} e^{6} x^{4}-35 b^{2} c \,e^{6} x^{4}+70 b \,c^{2} d \,e^{5} x^{4}-40 c^{3} d^{2} e^{4} x^{4}-210 a b c \,e^{6} x^{3}+280 a \,c^{2} d \,e^{5} x^{3}-35 b^{3} e^{6} x^{3}+280 b^{2} c d \,e^{5} x^{3}-560 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}+105 a^{2} c \,e^{6} x^{2}+105 a \,b^{2} e^{6} x^{2}-1260 a b c d \,e^{5} x^{2}+1680 a \,c^{2} d^{2} e^{4} x^{2}-210 b^{3} d \,e^{5} x^{2}+1680 b^{2} c \,d^{2} e^{4} x^{2}-3360 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+35 a^{2} b \,e^{6} x +140 a^{2} c d \,e^{5} x +140 a \,b^{2} d \,e^{5} x -1680 a b c \,d^{2} e^{4} x +2240 a \,c^{2} d^{3} e^{3} x -280 b^{3} d^{2} e^{4} x +2240 b^{2} c \,d^{3} e^{3} x -4480 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +7 a^{3} e^{6}+14 a^{2} b d \,e^{5}+56 a^{2} c \,d^{2} e^{4}+56 a \,b^{2} d^{2} e^{4}-672 a b c \,d^{3} e^{3}+896 a \,c^{2} d^{4} e^{2}-112 b^{3} d^{3} e^{3}+896 b^{2} c \,d^{4} e^{2}-1792 b \,c^{2} d^{5} e +1024 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{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)^(7/2),x)

[Out]

-2/35/(e*x+d)^(5/2)*(-5*c^3*e^6*x^6-21*b*c^2*e^6*x^5+12*c^3*d*e^5*x^5-35*a*c^2*e^6*x^4-35*b^2*c*e^6*x^4+70*b*c
^2*d*e^5*x^4-40*c^3*d^2*e^4*x^4-210*a*b*c*e^6*x^3+280*a*c^2*d*e^5*x^3-35*b^3*e^6*x^3+280*b^2*c*d*e^5*x^3-560*b
*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3+105*a^2*c*e^6*x^2+105*a*b^2*e^6*x^2-1260*a*b*c*d*e^5*x^2+1680*a*c^2*d^2*e
^4*x^2-210*b^3*d*e^5*x^2+1680*b^2*c*d^2*e^4*x^2-3360*b*c^2*d^3*e^3*x^2+1920*c^3*d^4*e^2*x^2+35*a^2*b*e^6*x+140
*a^2*c*d*e^5*x+140*a*b^2*d*e^5*x-1680*a*b*c*d^2*e^4*x+2240*a*c^2*d^3*e^3*x-280*b^3*d^2*e^4*x+2240*b^2*c*d^3*e^
3*x-4480*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+7*a^3*e^6+14*a^2*b*d*e^5+56*a^2*c*d^2*e^4+56*a*b^2*d^2*e^4-672*a*b*c
*d^3*e^3+896*a*c^2*d^4*e^2-112*b^3*d^3*e^3+896*b^2*c*d^4*e^2-1792*b*c^2*d^5*e+1024*c^3*d^6)/e^7

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maxima [A]  time = 0.90, size = 413, normalized size = 1.49 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\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 {3}{2}} - 35 \, {\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 )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\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} + 15 \, {\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 )} {\left (e x + d\right )}^{2} - 5 \, {\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 {5}{2}} e^{6}}\right )}}{35 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 35*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c +
a*c^2)*e^2)*(e*x + d)^(3/2) - 35*(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)*sqrt(e*x + d))/e^6 - 7*(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 + 15*(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)*(e*x + d)^2 - 5*(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)^(5/2)*e^6))/e

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mupad [B]  time = 0.89, size = 445, normalized size = 1.60 \begin {gather*} \frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\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 )}{3\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\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 )-\left (d+e\,x\right )\,\left (-2\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4-12\,a\,b\,c\,d^2\,e^3+8\,a\,c^2\,d^3\,e^2-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5\right )+\frac {2\,a^3\,e^6}{5}+\frac {2\,c^3\,d^6}{5}-\frac {2\,b^3\,d^3\,e^3}{5}+\frac {6\,a\,b^2\,d^2\,e^4}{5}+\frac {6\,a\,c^2\,d^4\,e^2}{5}+\frac {6\,a^2\,c\,d^2\,e^4}{5}+\frac {6\,b^2\,c\,d^4\,e^2}{5}-\frac {6\,a^2\,b\,d\,e^5}{5}-\frac {6\,b\,c^2\,d^5\,e}{5}-\frac {12\,a\,b\,c\,d^3\,e^3}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{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)^(7/2),x)

[Out]

(2*c^3*(d + e*x)^(7/2))/(7*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(5/2))/(5*e^7) + ((d + e*x)^(3/2)*(30*c^3*
d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(3*e^7) - ((d + e*x)^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) - (d + e*x)*(4*c^3*d^
5 - 2*a^2*b*e^5 - 2*b^3*d^2*e^3 + 8*a*c^2*d^3*e^2 + 8*b^2*c*d^3*e^2 + 4*a*b^2*d*e^4 + 4*a^2*c*d*e^4 - 10*b*c^2
*d^4*e - 12*a*b*c*d^2*e^3) + (2*a^3*e^6)/5 + (2*c^3*d^6)/5 - (2*b^3*d^3*e^3)/5 + (6*a*b^2*d^2*e^4)/5 + (6*a*c^
2*d^4*e^2)/5 + (6*a^2*c*d^2*e^4)/5 + (6*b^2*c*d^4*e^2)/5 - (6*a^2*b*d*e^5)/5 - (6*b*c^2*d^5*e)/5 - (12*a*b*c*d
^3*e^3)/5)/(e^7*(d + e*x)^(5/2)) + (2*(b*e - 2*c*d)*(d + e*x)^(1/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c
*d*e))/e^7

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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((c*x**2+b*x+a)**3/(e*x+d)**(7/2),x)

[Out]

Timed out

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