3.352 \(\int \frac{\left (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5} \]

[Out]

(-2*d^2*(c*d - b*e)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))
/(3*e^5*(d + e*x)^(3/2)) - (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2))/(e^5*Sqrt[d + e
*x]) - (4*c*(2*c*d - b*e)*Sqrt[d + e*x])/e^5 + (2*c^2*(d + e*x)^(3/2))/(3*e^5)

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Rubi [A]  time = 0.197401, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d^2*(c*d - b*e)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))
/(3*e^5*(d + e*x)^(3/2)) - (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2))/(e^5*Sqrt[d + e
*x]) - (4*c*(2*c*d - b*e)*Sqrt[d + e*x])/e^5 + (2*c^2*(d + e*x)^(3/2))/(3*e^5)

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Rubi in Sympy [A]  time = 32.8124, size = 138, normalized size = 0.97 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{5}} + \frac{4 c \sqrt{d + e x} \left (b e - 2 c d\right )}{e^{5}} - \frac{2 d^{2} \left (b e - c d\right )^{2}}{5 e^{5} \left (d + e x\right )^{\frac{5}{2}}} + \frac{4 d \left (b e - 2 c d\right ) \left (b e - c d\right )}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{e^{5} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**2/(e*x+d)**(7/2),x)

[Out]

2*c**2*(d + e*x)**(3/2)/(3*e**5) + 4*c*sqrt(d + e*x)*(b*e - 2*c*d)/e**5 - 2*d**2
*(b*e - c*d)**2/(5*e**5*(d + e*x)**(5/2)) + 4*d*(b*e - 2*c*d)*(b*e - c*d)/(3*e**
5*(d + e*x)**(3/2)) - 2*(b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(e**5*sqrt(d + e*x
))

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Mathematica [A]  time = 0.127428, size = 123, normalized size = 0.86 \[ -\frac{2 \left (b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30
*d*e^2*x^2 + 5*e^3*x^3) + c^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)))/(15*e^5*(d + e*x)^(5/2))

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Maple [A]  time = 0.01, size = 141, normalized size = 1. \[ -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}-60\,bc{e}^{4}{x}^{3}+80\,{c}^{2}d{e}^{3}{x}^{3}+30\,{b}^{2}{e}^{4}{x}^{2}-360\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+40\,{b}^{2}d{e}^{3}x-480\,bc{d}^{2}{e}^{2}x+640\,{c}^{2}{d}^{3}ex+16\,{b}^{2}{d}^{2}{e}^{2}-192\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2/(e*x+d)^(7/2),x)

[Out]

-2/15*(-5*c^2*e^4*x^4-30*b*c*e^4*x^3+40*c^2*d*e^3*x^3+15*b^2*e^4*x^2-180*b*c*d*e
^3*x^2+240*c^2*d^2*e^2*x^2+20*b^2*d*e^3*x-240*b*c*d^2*e^2*x+320*c^2*d^3*e*x+8*b^
2*d^2*e^2-96*b*c*d^3*e+128*c^2*d^4)/(e*x+d)^(5/2)/e^5

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Maxima [A]  time = 0.698758, size = 198, normalized size = 1.38 \[ \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 6 \,{\left (2 \, c^{2} d - b c e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} - 6 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 15 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 10 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((e*x + d)^(3/2)*c^2 - 6*(2*c^2*d - b*c*e)*sqrt(e*x + d))/e^4 - (3*c^2*d
^4 - 6*b*c*d^3*e + 3*b^2*d^2*e^2 + 15*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(e*x + d
)^2 - 10*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d))/((e*x + d)^(5/2)*e^4))
/e

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Fricas [A]  time = 0.221005, size = 212, normalized size = 1.48 \[ \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 8 \, b^{2} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} - 15 \,{\left (16 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 20 \,{\left (16 \, c^{2} d^{3} e - 12 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*c^2*e^4*x^4 - 128*c^2*d^4 + 96*b*c*d^3*e - 8*b^2*d^2*e^2 - 10*(4*c^2*d*e
^3 - 3*b*c*e^4)*x^3 - 15*(16*c^2*d^2*e^2 - 12*b*c*d*e^3 + b^2*e^4)*x^2 - 20*(16*
c^2*d^3*e - 12*b*c*d^2*e^2 + b^2*d*e^3)*x)/((e^7*x^2 + 2*d*e^6*x + d^2*e^5)*sqrt
(e*x + d))

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Sympy [A]  time = 10.0726, size = 787, normalized size = 5.5 \[ \begin{cases} - \frac{16 b^{2} d^{2} e^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{40 b^{2} d e^{3} x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{30 b^{2} e^{4} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{192 b c d^{3} e}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{480 b c d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{360 b c d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{60 b c e^{4} x^{3}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{\frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-16*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d +
e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 40*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d + e*
x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 30*b**2*e**4*x**2
/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d +
e*x)) + 192*b*c*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) + 480*b*c*d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x)
+ 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 360*b*c*d*e**3*x**2/
(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d +
e*x)) + 60*b*c*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) - 256*c**2*d**4/(15*d**2*e**5*sqrt(d + e*x) + 30*
d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*c**2*d**3*e*x/(15*d**
2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) -
480*c**2*d**2*e**2*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) - 80*c**2*d*e**3*x**3/(15*d**2*e**5*sqrt(d + e*x)
+ 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*c**2*e**4*x**4/(
15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e
*x)), Ne(e, 0)), ((b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5)/d**(7/2), True))

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GIAC/XCAS [A]  time = 0.208966, size = 242, normalized size = 1.69 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10} + 6 \, \sqrt{x e + d} b c e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \,{\left (x e + d\right )}^{2} b c d e + 30 \,{\left (x e + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \,{\left (x e + d\right )}^{2} b^{2} e^{2} - 10 \,{\left (x e + d\right )} b^{2} d e^{2} + 3 \, b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^2*e^10 - 12*sqrt(x*e + d)*c^2*d*e^10 + 6*sqrt(x*e + d)*b*
c*e^11)*e^(-15) - 2/15*(90*(x*e + d)^2*c^2*d^2 - 20*(x*e + d)*c^2*d^3 + 3*c^2*d^
4 - 90*(x*e + d)^2*b*c*d*e + 30*(x*e + d)*b*c*d^2*e - 6*b*c*d^3*e + 15*(x*e + d)
^2*b^2*e^2 - 10*(x*e + d)*b^2*d*e^2 + 3*b^2*d^2*e^2)*e^(-5)/(x*e + d)^(5/2)