3.1606 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx\)

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

[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)

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Rubi [A]  time = 0.3369, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{8 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac{4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac{2 c^2 (d+e x)^{5/2} (2 c d-b e)}{e^6}+\frac{4 c^3 (d+e x)^{7/2}}{7 e^6} \]

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)

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Rubi in Sympy [A]  time = 68.1971, size = 245, normalized size = 1. \[ \frac{4 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{6}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{e^{6}} + \frac{8 c \left (d + e x\right )^{\frac{3}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{e^{6}} - \frac{4 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6} \sqrt{d + e x}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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) + 2*c**2*(d + e*x)**(5/2)*(b*e - 2*c*d)/e**6 +
8*c*(d + e*x)**(3/2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(3*e**6) +
2*sqrt(d + e*x)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d*
*2)/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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Mathematica [A]  time = 0.325941, size = 289, normalized size = 1.17 \[ -\frac{2 \left (14 c e^2 \left (a^2 e^2 (2 d+3 e x)-3 a b e \left (8 d^2+12 d e x+3 e^2 x^2\right )+2 b^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+7 b e^3 \left (a^2 e^2+2 a b e (2 d+3 e x)+b^2 \left (-\left (8 d^2+12 d e x+3 e^2 x^2\right )\right )\right )-7 c^2 e \left (4 a e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+2 c^3 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{21 e^6 (d+e x)^{3/2}} \]

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 + 4
8*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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Maple [A]  time = 0.011, size = 359, normalized size = 1.5 \[ -{\frac{-12\,{c}^{3}{x}^{5}{e}^{5}-42\,b{c}^{2}{e}^{5}{x}^{4}+24\,{c}^{3}d{e}^{4}{x}^{4}-56\,a{c}^{2}{e}^{5}{x}^{3}-56\,{b}^{2}c{e}^{5}{x}^{3}+112\,b{c}^{2}d{e}^{4}{x}^{3}-64\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}-252\,abc{e}^{5}{x}^{2}+336\,a{c}^{2}d{e}^{4}{x}^{2}-42\,{b}^{3}{e}^{5}{x}^{2}+336\,{b}^{2}cd{e}^{4}{x}^{2}-672\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}+384\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+84\,{a}^{2}c{e}^{5}x+84\,a{b}^{2}{e}^{5}x-1008\,abcd{e}^{4}x+1344\,a{c}^{2}{d}^{2}{e}^{3}x-168\,{b}^{3}d{e}^{4}x+1344\,{b}^{2}c{d}^{2}{e}^{3}x-2688\,b{c}^{2}{d}^{3}{e}^{2}x+1536\,{c}^{3}{d}^{4}ex+14\,{a}^{2}b{e}^{5}+56\,{a}^{2}cd{e}^{4}+56\,a{b}^{2}d{e}^{4}-672\,abc{d}^{2}{e}^{3}+896\,a{c}^{2}{d}^{3}{e}^{2}-112\,{b}^{3}{d}^{2}{e}^{3}+896\,{b}^{2}c{d}^{3}{e}^{2}-1792\,b{c}^{2}{d}^{4}e+1024\,{c}^{3}{d}^{5}}{21\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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^4-336*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)/e^6

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Maxima [A]  time = 0.712368, size = 424, normalized size = 1.72 \[ \frac{2 \,{\left (\frac{6 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 21 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 28 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 21 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \sqrt{e x + d}}{e^{5}} + \frac{7 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 6 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{5}}\right )}}{21 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(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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Fricas [A]  time = 0.278018, size = 427, normalized size = 1.74 \[ \frac{2 \,{\left (6 \, c^{3} e^{5} x^{5} - 512 \, c^{3} d^{5} + 896 \, b c^{2} d^{4} e - 7 \, a^{2} b e^{5} - 448 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 56 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 28 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 3 \,{\left (4 \, c^{3} d e^{4} - 7 \, b c^{2} e^{5}\right )} x^{4} + 4 \,{\left (8 \, c^{3} d^{2} e^{3} - 14 \, b c^{2} d e^{4} + 7 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 3 \,{\left (64 \, c^{3} d^{3} e^{2} - 112 \, b c^{2} d^{2} e^{3} + 56 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} - 7 \,{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 6 \,{\left (128 \, c^{3} d^{4} e - 224 \, b c^{2} d^{3} e^{2} + 112 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 14 \,{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 7 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )}}{21 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(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)/((e^7*x
+ d*e^6)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{2}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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]

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

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GIAC/XCAS [A]  time = 0.280389, size = 594, normalized size = 2.41 \[ \frac{2}{21} \,{\left (6 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{36} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{36} + 140 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{36} - 420 \, \sqrt{x e + d} c^{3} d^{3} e^{36} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} e^{37} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d e^{37} + 630 \, \sqrt{x e + d} b c^{2} d^{2} e^{37} + 28 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c e^{38} + 28 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{38} - 252 \, \sqrt{x e + d} b^{2} c d e^{38} - 252 \, \sqrt{x e + d} a c^{2} d e^{38} + 21 \, \sqrt{x e + d} b^{3} e^{39} + 126 \, \sqrt{x e + d} a b c e^{39}\right )} e^{\left (-42\right )} - \frac{2 \,{\left (30 \,{\left (x e + d\right )} c^{3} d^{4} - 2 \, c^{3} d^{5} - 60 \,{\left (x e + d\right )} b c^{2} d^{3} e + 5 \, b c^{2} d^{4} e + 36 \,{\left (x e + d\right )} b^{2} c d^{2} e^{2} + 36 \,{\left (x e + d\right )} a c^{2} d^{2} e^{2} - 4 \, b^{2} c d^{3} e^{2} - 4 \, a c^{2} d^{3} e^{2} - 6 \,{\left (x e + d\right )} b^{3} d e^{3} - 36 \,{\left (x e + d\right )} a b c d e^{3} + b^{3} d^{2} e^{3} + 6 \, a b c d^{2} e^{3} + 6 \,{\left (x e + d\right )} a b^{2} e^{4} + 6 \,{\left (x e + d\right )} a^{2} c e^{4} - 2 \, a b^{2} d e^{4} - 2 \, a^{2} c d e^{4} + a^{2} b e^{5}\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(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^38 - 252*sqrt(x*e + d)*a*c^2*d*e^38 + 21*sqrt(x*e + d)*b^3*e^39 + 126*s
qrt(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)