3.359 \(\int \frac{(b x+c x^2)^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=244 \[ \frac{6 c (d+e x)^{5/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7}+\frac{6 d \sqrt{d+e x} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt{d+e x}}-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7} \]

[Out]

(-2*d^3*(c*d - b*e)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(e^7*Sqrt[d + e*x]) + (6*
d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^(3/2))/(3*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(5/2))/(5*e^7) - (6*c^
2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*x)^(9/2))/(9*e^7)

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Rubi [A]  time = 0.100583, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {698} \[ \frac{6 c (d+e x)^{5/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7}+\frac{6 d \sqrt{d+e x} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt{d+e x}}-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(-2*d^3*(c*d - b*e)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(e^7*Sqrt[d + e*x]) + (6*
d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^(3/2))/(3*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(5/2))/(5*e^7) - (6*c^
2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*x)^(9/2))/(9*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 (b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^{5/2}}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^{3/2}}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 \sqrt{d+e x}}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) \sqrt{d+e x}}{e^6}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{5/2}}{e^6}+\frac{c^3 (d+e x)^{7/2}}{e^6}\right ) \, dx\\ &=-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt{d+e x}}+\frac{6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{3 e^7}+\frac{6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}\\ \end{align*}

Mathematica [A]  time = 0.139984, size = 206, normalized size = 0.84 \[ \frac{2 \left (189 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-105 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+945 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-135 c^2 (d+e x)^5 (2 c d-b e)+945 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)-105 d^3 (c d-b e)^3+35 c^3 (d+e x)^6\right )}{315 e^7 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(2*(-105*d^3*(c*d - b*e)^3 + 945*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 945*d*(c*d - b*e)*(5*c^2*d^2 - 5*
b*c*d*e + b^2*e^2)*(d + e*x)^2 - 105*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 189*c*(5*
c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 135*c^2*(2*c*d - b*e)*(d + e*x)^5 + 35*c^3*(d + e*x)^6))/(315*e^7
*(d + e*x)^(3/2))

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Maple [A]  time = 0.048, size = 286, normalized size = 1.2 \begin{align*} -{\frac{-70\,{c}^{3}{x}^{6}{e}^{6}-270\,b{c}^{2}{e}^{6}{x}^{5}+120\,{c}^{3}d{e}^{5}{x}^{5}-378\,{b}^{2}c{e}^{6}{x}^{4}+540\,b{c}^{2}d{e}^{5}{x}^{4}-240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-210\,{b}^{3}{e}^{6}{x}^{3}+1008\,{b}^{2}cd{e}^{5}{x}^{3}-1440\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+1260\,{b}^{3}d{e}^{5}{x}^{2}-6048\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+8640\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+5040\,{b}^{3}{d}^{2}{e}^{4}x-24192\,{b}^{2}c{d}^{3}{e}^{3}x+34560\,b{c}^{2}{d}^{4}{e}^{2}x-15360\,{c}^{3}{d}^{5}ex+3360\,{b}^{3}{d}^{3}{e}^{3}-16128\,{b}^{2}c{d}^{4}{e}^{2}+23040\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{315\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^(5/2),x)

[Out]

-2/315*(-35*c^3*e^6*x^6-135*b*c^2*e^6*x^5+60*c^3*d*e^5*x^5-189*b^2*c*e^6*x^4+270*b*c^2*d*e^5*x^4-120*c^3*d^2*e
^4*x^4-105*b^3*e^6*x^3+504*b^2*c*d*e^5*x^3-720*b*c^2*d^2*e^4*x^3+320*c^3*d^3*e^3*x^3+630*b^3*d*e^5*x^2-3024*b^
2*c*d^2*e^4*x^2+4320*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2+2520*b^3*d^2*e^4*x-12096*b^2*c*d^3*e^3*x+17280*b*c
^2*d^4*e^2*x-7680*c^3*d^5*e*x+1680*b^3*d^3*e^3-8064*b^2*c*d^4*e^2+11520*b*c^2*d^5*e-5120*c^3*d^6)/(e*x+d)^(3/2
)/e^7

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Maxima [A]  time = 1.15751, size = 374, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 135 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 945 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{105 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} - 9 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{315 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*c^3 - 135*(2*c^3*d - b*c^2*e)*(e*x + d)^(7/2) + 189*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*
c*e^2)*(e*x + d)^(5/2) - 105*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(3/2) + 945*(5
*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*sqrt(e*x + d))/e^6 - 105*(c^3*d^6 - 3*b*c^2*d^5*e + 3
*b^2*c*d^4*e^2 - b^3*d^3*e^3 - 9*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d))/((e*x
+ d)^(3/2)*e^6))/e

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Fricas [A]  time = 2.4127, size = 645, normalized size = 2.64 \begin{align*} \frac{2 \,{\left (35 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 11520 \, b c^{2} d^{5} e + 8064 \, b^{2} c d^{4} e^{2} - 1680 \, b^{3} d^{3} e^{3} - 15 \,{\left (4 \, c^{3} d e^{5} - 9 \, b c^{2} e^{6}\right )} x^{5} + 3 \,{\left (40 \, c^{3} d^{2} e^{4} - 90 \, b c^{2} d e^{5} + 63 \, b^{2} c e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 720 \, b c^{2} d^{2} e^{4} + 504 \, b^{2} c d e^{5} - 105 \, b^{3} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{4} e^{2} - 720 \, b c^{2} d^{3} e^{3} + 504 \, b^{2} c d^{2} e^{4} - 105 \, b^{3} d e^{5}\right )} x^{2} + 24 \,{\left (320 \, c^{3} d^{5} e - 720 \, b c^{2} d^{4} e^{2} + 504 \, b^{2} c d^{3} e^{3} - 105 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*c^3*e^6*x^6 + 5120*c^3*d^6 - 11520*b*c^2*d^5*e + 8064*b^2*c*d^4*e^2 - 1680*b^3*d^3*e^3 - 15*(4*c^3*d
*e^5 - 9*b*c^2*e^6)*x^5 + 3*(40*c^3*d^2*e^4 - 90*b*c^2*d*e^5 + 63*b^2*c*e^6)*x^4 - (320*c^3*d^3*e^3 - 720*b*c^
2*d^2*e^4 + 504*b^2*c*d*e^5 - 105*b^3*e^6)*x^3 + 6*(320*c^3*d^4*e^2 - 720*b*c^2*d^3*e^3 + 504*b^2*c*d^2*e^4 -
105*b^3*d*e^5)*x^2 + 24*(320*c^3*d^5*e - 720*b*c^2*d^4*e^2 + 504*b^2*c*d^3*e^3 - 105*b^3*d^2*e^4)*x)*sqrt(e*x
+ d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 56.7304, size = 260, normalized size = 1.07 \begin{align*} \frac{2 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} + \frac{2 d^{3} \left (b e - c d\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} - \frac{6 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (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 )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (- 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 )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**(5/2),x)

[Out]

2*c**3*(d + e*x)**(9/2)/(9*e**7) + 2*d**3*(b*e - c*d)**3/(3*e**7*(d + e*x)**(3/2)) - 6*d**2*(b*e - 2*c*d)*(b*e
- c*d)**2/(e**7*sqrt(d + e*x)) + (d + e*x)**(7/2)*(6*b*c**2*e - 12*c**3*d)/(7*e**7) + (d + e*x)**(5/2)*(6*b**
2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(5*e**7) + (d + e*x)**(3/2)*(2*b**3*e**3 - 24*b**2*c*d*e**2 + 60*b*c*
*2*d**2*e - 40*c**3*d**3)/(3*e**7) + sqrt(d + e*x)*(-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)/e**7

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Giac [A]  time = 1.39004, size = 487, normalized size = 2. \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{56} - 270 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{56} + 945 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{56} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt{x e + d} c^{3} d^{4} e^{56} + 135 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{57} - 945 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{57} + 3150 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{57} - 9450 \, \sqrt{x e + d} b c^{2} d^{3} e^{57} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{58} + 5670 \, \sqrt{x e + d} b^{2} c d^{2} e^{58} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{59} - 945 \, \sqrt{x e + d} b^{3} d e^{59}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 45 \,{\left (x e + d\right )} b c^{2} d^{4} e + 3 \, b c^{2} d^{5} e + 36 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} - 3 \, b^{2} c d^{4} e^{2} - 9 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*e^56 - 270*(x*e + d)^(7/2)*c^3*d*e^56 + 945*(x*e + d)^(5/2)*c^3*d^2*e^56 - 2100*
(x*e + d)^(3/2)*c^3*d^3*e^56 + 4725*sqrt(x*e + d)*c^3*d^4*e^56 + 135*(x*e + d)^(7/2)*b*c^2*e^57 - 945*(x*e + d
)^(5/2)*b*c^2*d*e^57 + 3150*(x*e + d)^(3/2)*b*c^2*d^2*e^57 - 9450*sqrt(x*e + d)*b*c^2*d^3*e^57 + 189*(x*e + d)
^(5/2)*b^2*c*e^58 - 1260*(x*e + d)^(3/2)*b^2*c*d*e^58 + 5670*sqrt(x*e + d)*b^2*c*d^2*e^58 + 105*(x*e + d)^(3/2
)*b^3*e^59 - 945*sqrt(x*e + d)*b^3*d*e^59)*e^(-63) + 2/3*(18*(x*e + d)*c^3*d^5 - c^3*d^6 - 45*(x*e + d)*b*c^2*
d^4*e + 3*b*c^2*d^5*e + 36*(x*e + d)*b^2*c*d^3*e^2 - 3*b^2*c*d^4*e^2 - 9*(x*e + d)*b^3*d^2*e^3 + b^3*d^3*e^3)*
e^(-7)/(x*e + d)^(3/2)