3.2274 \(\int \frac{a+b x+c x^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2 \left (a e^2-b d e+c d^2\right )}{5 e^3 (d+e x)^{5/2}}+\frac{2 (2 c d-b e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 c}{e^3 \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.029924, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ -\frac{2 \left (a e^2-b d e+c d^2\right )}{5 e^3 (d+e x)^{5/2}}+\frac{2 (2 c d-b e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 c}{e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0490813, size = 54, normalized size = 0.74 \[ -\frac{2 \left (e (3 a e+2 b d+5 b e x)+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*(2*b*d + 3*a*e + 5*b*e*x) + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.042, size = 53, normalized size = 0.7 \begin{align*} -{\frac{30\,c{e}^{2}{x}^{2}+10\,b{e}^{2}x+40\,cdex+6\,a{e}^{2}+4\,bde+16\,c{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(15*c*e^2*x^2+5*b*e^2*x+20*c*d*e*x+3*a*e^2+2*b*d*e+8*c*d^2)/e^3

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Maxima [A]  time = 0.994225, size = 76, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c + 3 \, c d^{2} - 3 \, b d e + 3 \, a e^{2} - 5 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*(e*x + d)^2*c + 3*c*d^2 - 3*b*d*e + 3*a*e^2 - 5*(2*c*d - b*e)*(e*x + d))/((e*x + d)^(5/2)*e^3)

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Fricas [A]  time = 2.30313, size = 186, normalized size = 2.55 \begin{align*} -\frac{2 \,{\left (15 \, c e^{2} x^{2} + 8 \, c d^{2} + 2 \, b d e + 3 \, a e^{2} + 5 \,{\left (4 \, c d e + b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*c*e^2*x^2 + 8*c*d^2 + 2*b*d*e + 3*a*e^2 + 5*(4*c*d*e + b*e^2)*x)*sqrt(e*x + d)/(e^6*x^3 + 3*d*e^5*x^
2 + 3*d^2*e^4*x + d^3*e^3)

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Sympy [A]  time = 3.04277, size = 376, normalized size = 5.15 \begin{align*} \begin{cases} - \frac{6 a e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{4 b d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{10 b e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 4
*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 10*b*e**2*x/(15
*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*c*d**2/(15*d**2*e**3*s
qrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*c*d*e*x/(15*d**2*e**3*sqrt(d + e*x
) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*c*e**2*x**2/(15*d**2*e**3*sqrt(d + e*x) + 30*
d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a*x + b*x**2/2 + c*x**3/3)/d**(7/2), True))

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Giac [A]  time = 1.13541, size = 84, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} c - 10 \,{\left (x e + d\right )} c d + 3 \, c d^{2} + 5 \,{\left (x e + d\right )} b e - 3 \, b d e + 3 \, a e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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