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

Optimal. Leaf size=246 \[ \frac {8 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac {4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c^2 (d+e x)^{5/2} (2 c d-b e)}{e^6}+\frac {4 c^3 (d+e x)^{7/2}}{7 e^6} \]

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Rubi [A]  time = 0.13, antiderivative size = 246, 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 {8 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\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^6}-\frac {4 \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^6 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c^2 (d+e x)^{5/2} (2 c d-b e)}{e^6}+\frac {4 c^3 (d+e x)^{7/2}}{7 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^{5/2}}+\frac {2 \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^5 (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^5 \sqrt {d+e x}}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt {d+e x}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{3/2}}{e^5}+\frac {2 c^3 (d+e x)^{5/2}}{e^5}\right ) \, dx\\ &=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {4 \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^6 \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^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c^2 (2 c d-b e) (d+e x)^{5/2}}{e^6}+\frac {4 c^3 (d+e x)^{7/2}}{7 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 289, normalized size = 1.17 \begin {gather*} -\frac {2 \left (14 c e^2 \left (a^2 e^2 (2 d+3 e x)-3 a b e \left (8 d^2+12 d e x+3 e^2 x^2\right )+2 b^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+7 b e^3 \left (a^2 e^2+2 a b e (2 d+3 e x)-\left (b^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )\right )-7 c^2 e \left (4 a e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b \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 )+2 c^3 \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 )\right )}{21 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c^3*(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) + 7*b*e^3*(a^2*
e^2 + 2*a*b*e*(2*d + 3*e*x) - b^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2)) + 14*c*e^2*(a^2*e^2*(2*d + 3*e*x) - 3*a*b*e*
(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 2*b^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) - 7*c^2*e*(4*a*e*(-16*d^
3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b*(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)
)))/(21*e^6*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.17, size = 425, normalized size = 1.73 \begin {gather*} \frac {2 \left (-7 a^2 b e^5-42 a^2 c e^4 (d+e x)+14 a^2 c d e^4-42 a b^2 e^4 (d+e x)+14 a b^2 d e^4-42 a b c d^2 e^3+252 a b c d e^3 (d+e x)+126 a b c e^3 (d+e x)^2+28 a c^2 d^3 e^2-252 a c^2 d^2 e^2 (d+e x)-252 a c^2 d e^2 (d+e x)^2+28 a c^2 e^2 (d+e x)^3-7 b^3 d^2 e^3+42 b^3 d e^3 (d+e x)+21 b^3 e^3 (d+e x)^2+28 b^2 c d^3 e^2-252 b^2 c d^2 e^2 (d+e x)-252 b^2 c d e^2 (d+e x)^2+28 b^2 c e^2 (d+e x)^3-35 b c^2 d^4 e+420 b c^2 d^3 e (d+e x)+630 b c^2 d^2 e (d+e x)^2-140 b c^2 d e (d+e x)^3+21 b c^2 e (d+e x)^4+14 c^3 d^5-210 c^3 d^4 (d+e x)-420 c^3 d^3 (d+e x)^2+140 c^3 d^2 (d+e x)^3-42 c^3 d (d+e x)^4+6 c^3 (d+e x)^5\right )}{21 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(14*c^3*d^5 - 35*b*c^2*d^4*e + 28*b^2*c*d^3*e^2 + 28*a*c^2*d^3*e^2 - 7*b^3*d^2*e^3 - 42*a*b*c*d^2*e^3 + 14*
a*b^2*d*e^4 + 14*a^2*c*d*e^4 - 7*a^2*b*e^5 - 210*c^3*d^4*(d + e*x) + 420*b*c^2*d^3*e*(d + e*x) - 252*b^2*c*d^2
*e^2*(d + e*x) - 252*a*c^2*d^2*e^2*(d + e*x) + 42*b^3*d*e^3*(d + e*x) + 252*a*b*c*d*e^3*(d + e*x) - 42*a*b^2*e
^4*(d + e*x) - 42*a^2*c*e^4*(d + e*x) - 420*c^3*d^3*(d + e*x)^2 + 630*b*c^2*d^2*e*(d + e*x)^2 - 252*b^2*c*d*e^
2*(d + e*x)^2 - 252*a*c^2*d*e^2*(d + e*x)^2 + 21*b^3*e^3*(d + e*x)^2 + 126*a*b*c*e^3*(d + e*x)^2 + 140*c^3*d^2
*(d + e*x)^3 - 140*b*c^2*d*e*(d + e*x)^3 + 28*b^2*c*e^2*(d + e*x)^3 + 28*a*c^2*e^2*(d + e*x)^3 - 42*c^3*d*(d +
e*x)^4 + 21*b*c^2*e*(d + e*x)^4 + 6*c^3*(d + e*x)^5))/(21*e^6*(d + e*x)^(3/2))

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fricas [A]  time = 0.41, size = 327, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (6 \, c^{3} e^{5} x^{5} - 512 \, c^{3} d^{5} + 896 \, b c^{2} d^{4} e - 7 \, a^{2} b e^{5} - 448 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 56 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 28 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 3 \, {\left (4 \, c^{3} d e^{4} - 7 \, b c^{2} e^{5}\right )} x^{4} + 4 \, {\left (8 \, c^{3} d^{2} e^{3} - 14 \, b c^{2} d e^{4} + 7 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 3 \, {\left (64 \, c^{3} d^{3} e^{2} - 112 \, b c^{2} d^{2} e^{3} + 56 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - 7 \, {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 6 \, {\left (128 \, c^{3} d^{4} e - 224 \, b c^{2} d^{3} e^{2} + 112 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 14 \, {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 7 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(6*c^3*e^5*x^5 - 512*c^3*d^5 + 896*b*c^2*d^4*e - 7*a^2*b*e^5 - 448*(b^2*c + a*c^2)*d^3*e^2 + 56*(b^3 + 6*
a*b*c)*d^2*e^3 - 28*(a*b^2 + a^2*c)*d*e^4 - 3*(4*c^3*d*e^4 - 7*b*c^2*e^5)*x^4 + 4*(8*c^3*d^2*e^3 - 14*b*c^2*d*
e^4 + 7*(b^2*c + a*c^2)*e^5)*x^3 - 3*(64*c^3*d^3*e^2 - 112*b*c^2*d^2*e^3 + 56*(b^2*c + a*c^2)*d*e^4 - 7*(b^3 +
6*a*b*c)*e^5)*x^2 - 6*(128*c^3*d^4*e - 224*b*c^2*d^3*e^2 + 112*(b^2*c + a*c^2)*d^2*e^3 - 14*(b^3 + 6*a*b*c)*d
*e^4 + 7*(a*b^2 + a^2*c)*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)

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giac [A]  time = 0.22, size = 440, normalized size = 1.79 \begin {gather*} \frac {2}{21} \, {\left (6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} e^{36} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d e^{36} + 140 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{36} - 420 \, \sqrt {x e + d} c^{3} d^{3} e^{36} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} e^{37} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d e^{37} + 630 \, \sqrt {x e + d} b c^{2} d^{2} e^{37} + 28 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c e^{38} + 28 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} e^{38} - 252 \, \sqrt {x e + d} b^{2} c d e^{38} - 252 \, \sqrt {x e + d} a c^{2} d e^{38} + 21 \, \sqrt {x e + d} b^{3} e^{39} + 126 \, \sqrt {x e + d} a b c e^{39}\right )} e^{\left (-42\right )} - \frac {2 \, {\left (30 \, {\left (x e + d\right )} c^{3} d^{4} - 2 \, c^{3} d^{5} - 60 \, {\left (x e + d\right )} b c^{2} d^{3} e + 5 \, b c^{2} d^{4} e + 36 \, {\left (x e + d\right )} b^{2} c d^{2} e^{2} + 36 \, {\left (x e + d\right )} a c^{2} d^{2} e^{2} - 4 \, b^{2} c d^{3} e^{2} - 4 \, a c^{2} d^{3} e^{2} - 6 \, {\left (x e + d\right )} b^{3} d e^{3} - 36 \, {\left (x e + d\right )} a b c d e^{3} + b^{3} d^{2} e^{3} + 6 \, a b c d^{2} e^{3} + 6 \, {\left (x e + d\right )} a b^{2} e^{4} + 6 \, {\left (x e + d\right )} a^{2} c e^{4} - 2 \, a b^{2} d e^{4} - 2 \, a^{2} c d e^{4} + a^{2} b e^{5}\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(6*(x*e + d)^(7/2)*c^3*e^36 - 42*(x*e + d)^(5/2)*c^3*d*e^36 + 140*(x*e + d)^(3/2)*c^3*d^2*e^36 - 420*sqrt
(x*e + d)*c^3*d^3*e^36 + 21*(x*e + d)^(5/2)*b*c^2*e^37 - 140*(x*e + d)^(3/2)*b*c^2*d*e^37 + 630*sqrt(x*e + d)*
b*c^2*d^2*e^37 + 28*(x*e + d)^(3/2)*b^2*c*e^38 + 28*(x*e + d)^(3/2)*a*c^2*e^38 - 252*sqrt(x*e + d)*b^2*c*d*e^3
8 - 252*sqrt(x*e + d)*a*c^2*d*e^38 + 21*sqrt(x*e + d)*b^3*e^39 + 126*sqrt(x*e + d)*a*b*c*e^39)*e^(-42) - 2/3*(
30*(x*e + d)*c^3*d^4 - 2*c^3*d^5 - 60*(x*e + d)*b*c^2*d^3*e + 5*b*c^2*d^4*e + 36*(x*e + d)*b^2*c*d^2*e^2 + 36*
(x*e + d)*a*c^2*d^2*e^2 - 4*b^2*c*d^3*e^2 - 4*a*c^2*d^3*e^2 - 6*(x*e + d)*b^3*d*e^3 - 36*(x*e + d)*a*b*c*d*e^3
+ b^3*d^2*e^3 + 6*a*b*c*d^2*e^3 + 6*(x*e + d)*a*b^2*e^4 + 6*(x*e + d)*a^2*c*e^4 - 2*a*b^2*d*e^4 - 2*a^2*c*d*e
^4 + a^2*b*e^5)*e^(-6)/(x*e + d)^(3/2)

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maple [A]  time = 0.05, size = 359, normalized size = 1.46 \begin {gather*} -\frac {2 \left (-6 c^{3} e^{5} x^{5}-21 b \,c^{2} e^{5} x^{4}+12 c^{3} d \,e^{4} x^{4}-28 a \,c^{2} e^{5} x^{3}-28 b^{2} c \,e^{5} x^{3}+56 b \,c^{2} d \,e^{4} x^{3}-32 c^{3} d^{2} e^{3} x^{3}-126 a b c \,e^{5} x^{2}+168 a \,c^{2} d \,e^{4} x^{2}-21 b^{3} e^{5} x^{2}+168 b^{2} c d \,e^{4} x^{2}-336 b \,c^{2} d^{2} e^{3} x^{2}+192 c^{3} d^{3} e^{2} x^{2}+42 a^{2} c \,e^{5} x +42 a \,b^{2} e^{5} x -504 a b c d \,e^{4} x +672 a \,c^{2} d^{2} e^{3} x -84 b^{3} d \,e^{4} x +672 b^{2} c \,d^{2} e^{3} x -1344 b \,c^{2} d^{3} e^{2} x +768 c^{3} d^{4} e x +7 a^{2} b \,e^{5}+28 a^{2} c d \,e^{4}+28 a \,b^{2} d \,e^{4}-336 a b c \,d^{2} e^{3}+448 a \,c^{2} d^{3} e^{2}-56 b^{3} d^{2} e^{3}+448 b^{2} c \,d^{3} e^{2}-896 b \,c^{2} d^{4} e +512 c^{3} d^{5}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21/(e*x+d)^(3/2)*(-6*c^3*e^5*x^5-21*b*c^2*e^5*x^4+12*c^3*d*e^4*x^4-28*a*c^2*e^5*x^3-28*b^2*c*e^5*x^3+56*b*c
^2*d*e^4*x^3-32*c^3*d^2*e^3*x^3-126*a*b*c*e^5*x^2+168*a*c^2*d*e^4*x^2-21*b^3*e^5*x^2+168*b^2*c*d*e^4*x^2-336*b
*c^2*d^2*e^3*x^2+192*c^3*d^3*e^2*x^2+42*a^2*c*e^5*x+42*a*b^2*e^5*x-504*a*b*c*d*e^4*x+672*a*c^2*d^2*e^3*x-84*b^
3*d*e^4*x+672*b^2*c*d^2*e^3*x-1344*b*c^2*d^3*e^2*x+768*c^3*d^4*e*x+7*a^2*b*e^5+28*a^2*c*d*e^4+28*a*b^2*d*e^4-3
36*a*b*c*d^2*e^3+448*a*c^2*d^3*e^2-56*b^3*d^2*e^3+448*b^2*c*d^3*e^2-896*b*c^2*d^4*e+512*c^3*d^5)/e^6

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maxima [A]  time = 0.53, size = 314, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (\frac {6 \, {\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}} + 28 \, {\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}} - 21 \, {\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^{5}} + \frac {7 \, {\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} - 6 \, {\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 )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{21 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*((6*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 28*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c +
a*c^2)*e^2)*(e*x + d)^(3/2) - 21*(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^5 + 7*(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 - 6*(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))/((e*x + d)^(3/2)*e^5))/e

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mupad [B]  time = 1.89, size = 329, normalized size = 1.34 \begin {gather*} \frac {4\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{3\,e^6}+\frac {\frac {4\,c^3\,d^5}{3}-\left (d+e\,x\right )\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )-\frac {2\,a^2\,b\,e^5}{3}-\frac {2\,b^3\,d^2\,e^3}{3}+\frac {8\,a\,c^2\,d^3\,e^2}{3}+\frac {8\,b^2\,c\,d^3\,e^2}{3}+\frac {4\,a\,b^2\,d\,e^4}{3}+\frac {4\,a^2\,c\,d\,e^4}{3}-\frac {10\,b\,c^2\,d^4\,e}{3}-4\,a\,b\,c\,d^2\,e^3}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\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^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^3*(d + e*x)^(7/2))/(7*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d + e*x)^(5/2))/(5*e^6) + ((d + e*x)^(3/2)*(40*c^3
*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(3*e^6) + ((4*c^3*d^5)/3 - (d + e*x)*(20*c^3*d^4 + 4*a*b^2*e
^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 - 40*b*c^2*d^3*e - 24*a*b*c*d*e^3) - (2*a
^2*b*e^5)/3 - (2*b^3*d^2*e^3)/3 + (8*a*c^2*d^3*e^2)/3 + (8*b^2*c*d^3*e^2)/3 + (4*a*b^2*d*e^4)/3 + (4*a^2*c*d*e
^4)/3 - (10*b*c^2*d^4*e)/3 - 4*a*b*c*d^2*e^3)/(e^6*(d + e*x)^(3/2)) + (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^6

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sympy [A]  time = 113.80, size = 274, normalized size = 1.11 \begin {gather*} \frac {4 c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (10 b c^{2} e - 20 c^{3} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (8 a c^{2} e^{2} + 8 b^{2} c e^{2} - 40 b c^{2} d e + 40 c^{3} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \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 )}{e^{6}} - \frac {4 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6} \sqrt {d + e x}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*c**3*(d + e*x)**(7/2)/(7*e**6) + (d + e*x)**(5/2)*(10*b*c**2*e - 20*c**3*d)/(5*e**6) + (d + e*x)**(3/2)*(8*a
*c**2*e**2 + 8*b**2*c*e**2 - 40*b*c**2*d*e + 40*c**3*d**2)/(3*e**6) + sqrt(d + e*x)*(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)/e**6 - 4*(a*e**2 - b*d*e + c*d**2)
*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(e**6*sqrt(d + e*x)) - 2*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d
**2)**2/(3*e**6*(d + e*x)**(3/2))

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