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

Optimal. Leaf size=280 \[ \frac {6 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {6 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 280, 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)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {2 (d+e x)^{5/2} (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 )}{5 e^7}+\frac {2 (d+e x)^{3/2} \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 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.41, size = 394, normalized size = 1.41 \begin {gather*} \frac {-66 c e^2 \left (35 a^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 a b e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+3 b^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+462 e^3 \left (-5 a^3 e^3+15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left (-8 d^2-4 d e x+e^2 x^2\right )+b^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+22 c^2 e \left (9 a e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 b \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )-10 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x
^6) + 462*e^3*(-5*a^3*e^3 + 15*a^2*b*e^2*(2*d + e*x) + 5*a*b^2*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + b^3*(16*d^3 +
8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 66*c*e^2*(35*a^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*a*b*e*(16*d^3 + 8*
d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 3*b^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) +
22*c^2*e*(9*a*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*b*(256*d^5 + 128*d^4*e*
x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(1155*e^7*Sqrt[d + e*x])

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IntegrateAlgebraic [B]  time = 0.20, size = 592, normalized size = 2.11 \begin {gather*} \frac {2 \left (-1155 a^3 e^6+3465 a^2 b e^5 (d+e x)+3465 a^2 b d e^5-3465 a^2 c d^2 e^4-6930 a^2 c d e^4 (d+e x)+1155 a^2 c e^4 (d+e x)^2-3465 a b^2 d^2 e^4-6930 a b^2 d e^4 (d+e x)+1155 a b^2 e^4 (d+e x)^2+6930 a b c d^3 e^3+20790 a b c d^2 e^3 (d+e x)-6930 a b c d e^3 (d+e x)^2+1386 a b c e^3 (d+e x)^3-3465 a c^2 d^4 e^2-13860 a c^2 d^3 e^2 (d+e x)+6930 a c^2 d^2 e^2 (d+e x)^2-2772 a c^2 d e^2 (d+e x)^3+495 a c^2 e^2 (d+e x)^4+1155 b^3 d^3 e^3+3465 b^3 d^2 e^3 (d+e x)-1155 b^3 d e^3 (d+e x)^2+231 b^3 e^3 (d+e x)^3-3465 b^2 c d^4 e^2-13860 b^2 c d^3 e^2 (d+e x)+6930 b^2 c d^2 e^2 (d+e x)^2-2772 b^2 c d e^2 (d+e x)^3+495 b^2 c e^2 (d+e x)^4+3465 b c^2 d^5 e+17325 b c^2 d^4 e (d+e x)-11550 b c^2 d^3 e (d+e x)^2+6930 b c^2 d^2 e (d+e x)^3-2475 b c^2 d e (d+e x)^4+385 b c^2 e (d+e x)^5-1155 c^3 d^6-6930 c^3 d^5 (d+e x)+5775 c^3 d^4 (d+e x)^2-4620 c^3 d^3 (d+e x)^3+2475 c^3 d^2 (d+e x)^4-770 c^3 d (d+e x)^5+105 c^3 (d+e x)^6\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*c^3*d^6 + 3465*b*c^2*d^5*e - 3465*b^2*c*d^4*e^2 - 3465*a*c^2*d^4*e^2 + 1155*b^3*d^3*e^3 + 6930*a*b*c
*d^3*e^3 - 3465*a*b^2*d^2*e^4 - 3465*a^2*c*d^2*e^4 + 3465*a^2*b*d*e^5 - 1155*a^3*e^6 - 6930*c^3*d^5*(d + e*x)
+ 17325*b*c^2*d^4*e*(d + e*x) - 13860*b^2*c*d^3*e^2*(d + e*x) - 13860*a*c^2*d^3*e^2*(d + e*x) + 3465*b^3*d^2*e
^3*(d + e*x) + 20790*a*b*c*d^2*e^3*(d + e*x) - 6930*a*b^2*d*e^4*(d + e*x) - 6930*a^2*c*d*e^4*(d + e*x) + 3465*
a^2*b*e^5*(d + e*x) + 5775*c^3*d^4*(d + e*x)^2 - 11550*b*c^2*d^3*e*(d + e*x)^2 + 6930*b^2*c*d^2*e^2*(d + e*x)^
2 + 6930*a*c^2*d^2*e^2*(d + e*x)^2 - 1155*b^3*d*e^3*(d + e*x)^2 - 6930*a*b*c*d*e^3*(d + e*x)^2 + 1155*a*b^2*e^
4*(d + e*x)^2 + 1155*a^2*c*e^4*(d + e*x)^2 - 4620*c^3*d^3*(d + e*x)^3 + 6930*b*c^2*d^2*e*(d + e*x)^3 - 2772*b^
2*c*d*e^2*(d + e*x)^3 - 2772*a*c^2*d*e^2*(d + e*x)^3 + 231*b^3*e^3*(d + e*x)^3 + 1386*a*b*c*e^3*(d + e*x)^3 +
2475*c^3*d^2*(d + e*x)^4 - 2475*b*c^2*d*e*(d + e*x)^4 + 495*b^2*c*e^2*(d + e*x)^4 + 495*a*c^2*e^2*(d + e*x)^4
- 770*c^3*d*(d + e*x)^5 + 385*b*c^2*e*(d + e*x)^5 + 105*c^3*(d + e*x)^6))/(1155*e^7*Sqrt[d + e*x])

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fricas [A]  time = 0.41, size = 417, normalized size = 1.49 \begin {gather*} \frac {2 \, {\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e + 6930 \, a^{2} b d e^{5} - 1155 \, a^{3} e^{6} - 12672 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3696 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 9240 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 35 \, {\left (4 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (40 \, c^{3} d^{2} e^{4} - 110 \, b c^{2} d e^{5} + 99 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - {\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 231 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + {\left (640 \, c^{3} d^{4} e^{2} - 1760 \, b c^{2} d^{3} e^{3} + 1584 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 462 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 1155 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{5} e - 7040 \, b c^{2} d^{4} e^{2} - 3465 \, a^{2} b e^{6} + 6336 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1848 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4620 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{1155 \, {\left (e^{8} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(105*c^3*e^6*x^6 - 5120*c^3*d^6 + 14080*b*c^2*d^5*e + 6930*a^2*b*d*e^5 - 1155*a^3*e^6 - 12672*(b^2*c +
a*c^2)*d^4*e^2 + 3696*(b^3 + 6*a*b*c)*d^3*e^3 - 9240*(a*b^2 + a^2*c)*d^2*e^4 - 35*(4*c^3*d*e^5 - 11*b*c^2*e^6)
*x^5 + 5*(40*c^3*d^2*e^4 - 110*b*c^2*d*e^5 + 99*(b^2*c + a*c^2)*e^6)*x^4 - (320*c^3*d^3*e^3 - 880*b*c^2*d^2*e^
4 + 792*(b^2*c + a*c^2)*d*e^5 - 231*(b^3 + 6*a*b*c)*e^6)*x^3 + (640*c^3*d^4*e^2 - 1760*b*c^2*d^3*e^3 + 1584*(b
^2*c + a*c^2)*d^2*e^4 - 462*(b^3 + 6*a*b*c)*d*e^5 + 1155*(a*b^2 + a^2*c)*e^6)*x^2 - (2560*c^3*d^5*e - 7040*b*c
^2*d^4*e^2 - 3465*a^2*b*e^6 + 6336*(b^2*c + a*c^2)*d^3*e^3 - 1848*(b^3 + 6*a*b*c)*d^2*e^4 + 4620*(a*b^2 + a^2*
c)*d*e^5)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)

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giac [B]  time = 0.24, size = 631, normalized size = 2.25 \begin {gather*} \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} e^{70} - 770 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d e^{70} + 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{2} e^{70} - 4620 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{70} + 5775 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt {x e + d} c^{3} d^{5} e^{70} + 385 \, {\left (x e + d\right )}^{\frac {9}{2}} b c^{2} e^{71} - 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} d e^{71} + 6930 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d^{2} e^{71} - 11550 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{3} e^{71} + 17325 \, \sqrt {x e + d} b c^{2} d^{4} e^{71} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c e^{72} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} e^{72} - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c d e^{72} - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d e^{72} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d^{2} e^{72} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{2} e^{72} - 13860 \, \sqrt {x e + d} b^{2} c d^{3} e^{72} - 13860 \, \sqrt {x e + d} a c^{2} d^{3} e^{72} + 231 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{73} + 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} a b c e^{73} - 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{73} - 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c d e^{73} + 3465 \, \sqrt {x e + d} b^{3} d^{2} e^{73} + 20790 \, \sqrt {x e + d} a b c d^{2} e^{73} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} e^{74} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c e^{74} - 6930 \, \sqrt {x e + d} a b^{2} d e^{74} - 6930 \, \sqrt {x e + d} a^{2} c d e^{74} + 3465 \, \sqrt {x e + d} a^{2} b e^{75}\right )} e^{\left (-77\right )} - \frac {2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*e^70 - 770*(x*e + d)^(9/2)*c^3*d*e^70 + 2475*(x*e + d)^(7/2)*c^3*d^2*e^70 - 4
620*(x*e + d)^(5/2)*c^3*d^3*e^70 + 5775*(x*e + d)^(3/2)*c^3*d^4*e^70 - 6930*sqrt(x*e + d)*c^3*d^5*e^70 + 385*(
x*e + d)^(9/2)*b*c^2*e^71 - 2475*(x*e + d)^(7/2)*b*c^2*d*e^71 + 6930*(x*e + d)^(5/2)*b*c^2*d^2*e^71 - 11550*(x
*e + d)^(3/2)*b*c^2*d^3*e^71 + 17325*sqrt(x*e + d)*b*c^2*d^4*e^71 + 495*(x*e + d)^(7/2)*b^2*c*e^72 + 495*(x*e
+ d)^(7/2)*a*c^2*e^72 - 2772*(x*e + d)^(5/2)*b^2*c*d*e^72 - 2772*(x*e + d)^(5/2)*a*c^2*d*e^72 + 6930*(x*e + d)
^(3/2)*b^2*c*d^2*e^72 + 6930*(x*e + d)^(3/2)*a*c^2*d^2*e^72 - 13860*sqrt(x*e + d)*b^2*c*d^3*e^72 - 13860*sqrt(
x*e + d)*a*c^2*d^3*e^72 + 231*(x*e + d)^(5/2)*b^3*e^73 + 1386*(x*e + d)^(5/2)*a*b*c*e^73 - 1155*(x*e + d)^(3/2
)*b^3*d*e^73 - 6930*(x*e + d)^(3/2)*a*b*c*d*e^73 + 3465*sqrt(x*e + d)*b^3*d^2*e^73 + 20790*sqrt(x*e + d)*a*b*c
*d^2*e^73 + 1155*(x*e + d)^(3/2)*a*b^2*e^74 + 1155*(x*e + d)^(3/2)*a^2*c*e^74 - 6930*sqrt(x*e + d)*a*b^2*d*e^7
4 - 6930*sqrt(x*e + d)*a^2*c*d*e^74 + 3465*sqrt(x*e + d)*a^2*b*e^75)*e^(-77) - 2*(c^3*d^6 - 3*b*c^2*d^5*e + 3*
b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*
d*e^5 + a^3*e^6)*e^(-7)/sqrt(x*e + d)

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maple [A]  time = 0.05, size = 495, normalized size = 1.77 \begin {gather*} -\frac {2 \left (-105 c^{3} x^{6} e^{6}-385 b \,c^{2} e^{6} x^{5}+140 c^{3} d \,e^{5} x^{5}-495 a \,c^{2} e^{6} x^{4}-495 b^{2} c \,e^{6} x^{4}+550 b \,c^{2} d \,e^{5} x^{4}-200 c^{3} d^{2} e^{4} x^{4}-1386 a b c \,e^{6} x^{3}+792 a \,c^{2} d \,e^{5} x^{3}-231 b^{3} e^{6} x^{3}+792 b^{2} c d \,e^{5} x^{3}-880 b \,c^{2} d^{2} e^{4} x^{3}+320 c^{3} d^{3} e^{3} x^{3}-1155 a^{2} c \,e^{6} x^{2}-1155 a \,b^{2} e^{6} x^{2}+2772 a b c d \,e^{5} x^{2}-1584 a \,c^{2} d^{2} e^{4} x^{2}+462 b^{3} d \,e^{5} x^{2}-1584 b^{2} c \,d^{2} e^{4} x^{2}+1760 b \,c^{2} d^{3} e^{3} x^{2}-640 c^{3} d^{4} e^{2} x^{2}-3465 a^{2} b \,e^{6} x +4620 a^{2} c d \,e^{5} x +4620 a \,b^{2} d \,e^{5} x -11088 a b c \,d^{2} e^{4} x +6336 a \,c^{2} d^{3} e^{3} x -1848 b^{3} d^{2} e^{4} x +6336 b^{2} c \,d^{3} e^{3} x -7040 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +1155 a^{3} e^{6}-6930 a^{2} b d \,e^{5}+9240 a^{2} c \,d^{2} e^{4}+9240 a \,b^{2} d^{2} e^{4}-22176 a b c \,d^{3} e^{3}+12672 a \,c^{2} d^{4} e^{2}-3696 b^{3} d^{3} e^{3}+12672 b^{2} c \,d^{4} e^{2}-14080 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{1155 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/1155/(e*x+d)^(1/2)*(-105*c^3*e^6*x^6-385*b*c^2*e^6*x^5+140*c^3*d*e^5*x^5-495*a*c^2*e^6*x^4-495*b^2*c*e^6*x^
4+550*b*c^2*d*e^5*x^4-200*c^3*d^2*e^4*x^4-1386*a*b*c*e^6*x^3+792*a*c^2*d*e^5*x^3-231*b^3*e^6*x^3+792*b^2*c*d*e
^5*x^3-880*b*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3-1155*a^2*c*e^6*x^2-1155*a*b^2*e^6*x^2+2772*a*b*c*d*e^5*x^2-15
84*a*c^2*d^2*e^4*x^2+462*b^3*d*e^5*x^2-1584*b^2*c*d^2*e^4*x^2+1760*b*c^2*d^3*e^3*x^2-640*c^3*d^4*e^2*x^2-3465*
a^2*b*e^6*x+4620*a^2*c*d*e^5*x+4620*a*b^2*d*e^5*x-11088*a*b*c*d^2*e^4*x+6336*a*c^2*d^3*e^3*x-1848*b^3*d^2*e^4*
x+6336*b^2*c*d^3*e^3*x-7040*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+1155*a^3*e^6-6930*a^2*b*d*e^5+9240*a^2*c*d^2*e^4+
9240*a*b^2*d^2*e^4-22176*a*b*c*d^3*e^3+12672*a*c^2*d^4*e^2-3696*b^3*d^3*e^3+12672*b^2*c*d^4*e^2-14080*b*c^2*d^
5*e+5120*c^3*d^6)/e^7

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maxima [A]  time = 0.95, size = 415, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (\frac {105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} - 385 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\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 {7}{2}} - 231 \, {\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 {5}{2}} + 1155 \, {\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 )}^{\frac {3}{2}} - 3465 \, {\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 )} \sqrt {e x + d}}{e^{6}} - \frac {1155 \, {\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}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{1155 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/1155*((105*(e*x + d)^(11/2)*c^3 - 385*(2*c^3*d - b*c^2*e)*(e*x + d)^(9/2) + 495*(5*c^3*d^2 - 5*b*c^2*d*e + (
b^2*c + a*c^2)*e^2)*(e*x + d)^(7/2) - 231*(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)^(5/2) + 1155*(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)^(3/2) - 3465*(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)*sqrt(e*x + d))/e^6 - 1155*(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)/
(sqrt(e*x + d)*e^6))/e

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mupad [B]  time = 0.87, size = 386, normalized size = 1.38 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\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 )}{3\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\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 )}{7\,e^7}-\frac {2\,a^3\,e^6-6\,a^2\,b\,d\,e^5+6\,a^2\,c\,d^2\,e^4+6\,a\,b^2\,d^2\,e^4-12\,a\,b\,c\,d^3\,e^3+6\,a\,c^2\,d^4\,e^2-2\,b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2-6\,b\,c^2\,d^5\,e+2\,c^3\,d^6}{e^7\,\sqrt {d+e\,x}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{5\,e^7}+\frac {6\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{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)^(3/2),x)

[Out]

((d + e*x)^(3/2)*(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))/(3*e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d +
e*x)^(9/2))/(9*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(7*e^7) - (2*a
^3*e^6 + 2*c^3*d^6 - 2*b^3*d^3*e^3 + 6*a*b^2*d^2*e^4 + 6*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 + 6*b^2*c*d^4*e^2 - 6
*a^2*b*d*e^5 - 6*b*c^2*d^5*e - 12*a*b*c*d^3*e^3)/(e^7*(d + e*x)^(1/2)) + (2*(b*e - 2*c*d)*(d + e*x)^(5/2)*(b^2
*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(5*e^7) + (6*(b*e - 2*c*d)*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e
)^2)/e^7

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sympy [A]  time = 138.86, size = 428, normalized size = 1.53 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{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 )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{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 )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{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 )}{3 e^{7}} + \frac {\sqrt {d + e x} \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 )}{e^{7}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{7} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**3*(d + e*x)**(11/2)/(11*e**7) + (d + e*x)**(9/2)*(6*b*c**2*e - 12*c**3*d)/(9*e**7) + (d + e*x)**(7/2)*(6*
a*c**2*e**2 + 6*b**2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**(3/2)*(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 - 6
0*b*c**2*d**3*e + 30*c**3*d**4)/(3*e**7) + sqrt(d + e*x)*(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)/e**7 - 2*(a*e**2 - b*d*e + c*d**2)**3/(e**7*sqrt(d + e*x))

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