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

Optimal. Leaf size=128 \[ -\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 c \sqrt {d+e x} (2 c d-b e)}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4} \]

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Rubi [A]  time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 c \sqrt {d+e x} (2 c d-b e)}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.12, size = 108, normalized size = 0.84 \begin {gather*} -\frac {2 \left (c e \left (2 a e (2 d+3 e x)-3 b \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+b e^2 (a e+2 b d+3 b e x)+2 c^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.08, size = 142, normalized size = 1.11 \begin {gather*} \frac {2 \left (-a b e^3-6 a c e^2 (d+e x)+2 a c d e^2-3 b^2 e^2 (d+e x)+b^2 d e^2-3 b c d^2 e+18 b c d e (d+e x)+9 b c e (d+e x)^2+2 c^2 d^3-18 c^2 d^2 (d+e x)-18 c^2 d (d+e x)^2+2 c^2 (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 - 18*c^2*d^2*(d + e*x) + 18*b*c*d*e*(d + e*x)
- 3*b^2*e^2*(d + e*x) - 6*a*c*e^2*(d + e*x) - 18*c^2*d*(d + e*x)^2 + 9*b*c*e*(d + e*x)^2 + 2*c^2*(d + e*x)^3))
/(3*e^4*(d + e*x)^(3/2))

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fricas [A]  time = 0.41, size = 137, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (2 \, c^{2} e^{3} x^{3} - 32 \, c^{2} d^{3} + 24 \, b c d^{2} e - a b e^{3} - 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (4 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} - 3 \, {\left (16 \, c^{2} d^{2} e - 12 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\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)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/3*(2*c^2*e^3*x^3 - 32*c^2*d^3 + 24*b*c*d^2*e - a*b*e^3 - 2*(b^2 + 2*a*c)*d*e^2 - 3*(4*c^2*d*e^2 - 3*b*c*e^3)
*x^2 - 3*(16*c^2*d^2*e - 12*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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giac [A]  time = 0.20, size = 153, normalized size = 1.20 \begin {gather*} \frac {2}{3} \, {\left (2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} e^{8} - 18 \, \sqrt {x e + d} c^{2} d e^{8} + 9 \, \sqrt {x e + d} b c e^{9}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (18 \, {\left (x e + d\right )} c^{2} d^{2} - 2 \, c^{2} d^{3} - 18 \, {\left (x e + d\right )} b c d e + 3 \, b c d^{2} e + 3 \, {\left (x e + d\right )} b^{2} e^{2} + 6 \, {\left (x e + d\right )} a c e^{2} - b^{2} d e^{2} - 2 \, a c d e^{2} + a b e^{3}\right )} e^{\left (-4\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)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3*(2*(x*e + d)^(3/2)*c^2*e^8 - 18*sqrt(x*e + d)*c^2*d*e^8 + 9*sqrt(x*e + d)*b*c*e^9)*e^(-12) - 2/3*(18*(x*e
+ d)*c^2*d^2 - 2*c^2*d^3 - 18*(x*e + d)*b*c*d*e + 3*b*c*d^2*e + 3*(x*e + d)*b^2*e^2 + 6*(x*e + d)*a*c*e^2 - b^
2*d*e^2 - 2*a*c*d*e^2 + a*b*e^3)*e^(-4)/(x*e + d)^(3/2)

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maple [A]  time = 0.04, size = 122, normalized size = 0.95 \begin {gather*} -\frac {2 \left (-2 c^{2} x^{3} e^{3}-9 b c \,e^{3} x^{2}+12 c^{2} d \,e^{2} x^{2}+6 a c \,e^{3} x +3 b^{2} e^{3} x -36 b c d \,e^{2} x +48 c^{2} d^{2} e x +a b \,e^{3}+4 a c d \,e^{2}+2 b^{2} d \,e^{2}-24 b c \,d^{2} e +32 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(e*x+d)^(3/2)*(-2*c^2*e^3*x^3-9*b*c*e^3*x^2+12*c^2*d*e^2*x^2+6*a*c*e^3*x+3*b^2*e^3*x-36*b*c*d*e^2*x+48*c^
2*d^2*e*x+a*b*e^3+4*a*c*d*e^2+2*b^2*d*e^2-24*b*c*d^2*e+32*c^2*d^3)/e^4

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maxima [A]  time = 0.60, size = 126, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (\frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} - 9 \, {\left (2 \, c^{2} d - b c e\right )} \sqrt {e x + d}}{e^{3}} + \frac {2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, 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)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/3*((2*(e*x + d)^(3/2)*c^2 - 9*(2*c^2*d - b*c*e)*sqrt(e*x + d))/e^3 + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b
^2 + 2*a*c)*d*e^2 - 3*(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e

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mupad [B]  time = 0.09, size = 139, normalized size = 1.09 \begin {gather*} \frac {4\,c^2\,{\left (d+e\,x\right )}^3+4\,c^2\,d^3+2\,b^2\,d\,e^2-6\,b^2\,e^2\,\left (d+e\,x\right )-36\,c^2\,d\,{\left (d+e\,x\right )}^2-36\,c^2\,d^2\,\left (d+e\,x\right )-2\,a\,b\,e^3+4\,a\,c\,d\,e^2-6\,b\,c\,d^2\,e-12\,a\,c\,e^2\,\left (d+e\,x\right )+18\,b\,c\,e\,{\left (d+e\,x\right )}^2+36\,b\,c\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^2*(d + e*x)^3 + 4*c^2*d^3 + 2*b^2*d*e^2 - 6*b^2*e^2*(d + e*x) - 36*c^2*d*(d + e*x)^2 - 36*c^2*d^2*(d + e*
x) - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e - 12*a*c*e^2*(d + e*x) + 18*b*c*e*(d + e*x)^2 + 36*b*c*d*e*(d + e*x
))/(3*e^4*(d + e*x)^(3/2))

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sympy [A]  time = 1.65, size = 536, normalized size = 4.19 \begin {gather*} \begin {cases} - \frac {2 a b e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {8 a c d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a c e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 b^{2} d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 b^{2} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 b c d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 b c d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 b c e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {64 c^{2} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {96 c^{2} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {24 c^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {4 c^{2} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a b x + a c x^{2} + \frac {b^{2} x^{2}}{2} + b c x^{3} + \frac {c^{2} x^{4}}{2}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a*b*e**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 8*a*c*d*e**2/(3*d*e**4*sqrt(d + e*x
) + 3*e**5*x*sqrt(d + e*x)) - 12*a*c*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 4*b**2*d*e**2/
(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*b**2*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d +
e*x)) + 48*b*c*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 72*b*c*d*e**2*x/(3*d*e**4*sqrt(d + e
*x) + 3*e**5*x*sqrt(d + e*x)) + 18*b*c*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 64*c**2*d
**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 96*c**2*d**2*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sq
rt(d + e*x)) - 24*c**2*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 4*c**2*e**3*x**3/(3*d*e
**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), ((a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**
4/2)/d**(5/2), True))

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