3.20 \(\int \frac{(b x+c x^2)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(7*b*x^6) + (4*c*(b*x + c*x^2)^(5/2))/(35*b^2*x^5)

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Rubi [A]  time = 0.0164416, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(3/2)/x^6,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(7*b*x^6) + (4*c*(b*x + c*x^2)^(5/2))/(35*b^2*x^5)

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}-\frac{(2 c) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{7 b}\\ &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}+\frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0120312, size = 29, normalized size = 0.6 \[ \frac{2 (x (b+c x))^{5/2} (2 c x-5 b)}{35 b^2 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(3/2)/x^6,x]

[Out]

(2*(x*(b + c*x))^(5/2)*(-5*b + 2*c*x))/(35*b^2*x^6)

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Maple [A]  time = 0.051, size = 33, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+5\,b \right ) }{35\,{x}^{5}{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(3/2)/x^6,x)

[Out]

-2/35*(c*x+b)*(-2*c*x+5*b)*(c*x^2+b*x)^(3/2)/x^5/b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.06845, size = 105, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (2 \, c^{3} x^{3} - b c^{2} x^{2} - 8 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, b^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^6,x, algorithm="fricas")

[Out]

2/35*(2*c^3*x^3 - b*c^2*x^2 - 8*b^2*c*x - 5*b^3)*sqrt(c*x^2 + b*x)/(b^2*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(3/2)/x**6,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**6, x)

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Giac [B]  time = 1.25156, size = 223, normalized size = 4.65 \begin{align*} \frac{2 \,{\left (35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b c^{2} + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} c^{\frac{3}{2}} + 98 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{3} c + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{4} \sqrt{c} + 5 \, b^{5}\right )}}{35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^6,x, algorithm="giac")

[Out]

2/35*(35*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*c^(5/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b*c^2 + 140*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*b^2*c^(3/2) + 98*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c + 35*(sqrt(c)*x - sqrt(c*x
^2 + b*x))*b^4*sqrt(c) + 5*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^7