3.90 $$\int \frac{(b x+c x^2)^{3/2}}{x^{9/2}} \, dx$$

Optimal. Leaf size=83 $-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}$

[Out]

(-3*c*Sqrt[b*x + c*x^2])/(4*x^(3/2)) - (b*x + c*x^2)^(3/2)/(2*x^(7/2)) - (3*c^2*ArcTanh[Sqrt[b*x + c*x^2]/(Sqr
t[b]*Sqrt[x])])/(4*Sqrt[b])

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Rubi [A]  time = 0.0323445, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.158, Rules used = {662, 660, 207} $-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^(3/2)/x^(9/2),x]

[Out]

(-3*c*Sqrt[b*x + c*x^2])/(4*x^(3/2)) - (b*x + c*x^2)^(3/2)/(2*x^(7/2)) - (3*c^2*ArcTanh[Sqrt[b*x + c*x^2]/(Sqr
t[b]*Sqrt[x])])/(4*Sqrt[b])

Rule 662

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

Rule 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx &=-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac{1}{4} (3 c) \int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx\\ &=-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac{1}{8} \left (3 c^2\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac{1}{4} \left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0491958, size = 72, normalized size = 0.87 $-\frac{2 b^2+3 c^2 x^2 \sqrt{\frac{c x}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )+7 b c x+5 c^2 x^2}{4 x^{3/2} \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^(3/2)/x^(9/2),x]

[Out]

-(2*b^2 + 7*b*c*x + 5*c^2*x^2 + 3*c^2*x^2*Sqrt[1 + (c*x)/b]*ArcTanh[Sqrt[1 + (c*x)/b]])/(4*x^(3/2)*Sqrt[x*(b +
c*x)])

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Maple [A]  time = 0.192, size = 72, normalized size = 0.9 \begin{align*} -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}+5\,xc\sqrt{cx+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(3/2)/x^(9/2),x)

[Out]

-1/4*(x*(c*x+b))^(1/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*x^2*c^2+5*x*c*(c*x+b)^(1/2)*b^(1/2)+2*b^(3/2)*(c*x+b)
^(1/2))/x^(5/2)/(c*x+b)^(1/2)/b^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{x^{\frac{9}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(3/2)/x^(9/2), x)

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Fricas [A]  time = 2.19315, size = 367, normalized size = 4.42 \begin{align*} \left [\frac{3 \, \sqrt{b} c^{2} x^{3} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) - 2 \,{\left (5 \, b c x + 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b x^{3}}, \frac{3 \, \sqrt{-b} c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (5 \, b c x + 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b x^{3}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^(9/2),x, algorithm="fricas")

[Out]

[1/8*(3*sqrt(b)*c^2*x^3*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) - 2*(5*b*c*x + 2*b^2)*
sqrt(c*x^2 + b*x)*sqrt(x))/(b*x^3), 1/4*(3*sqrt(-b)*c^2*x^3*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (5*b*
c*x + 2*b^2)*sqrt(c*x^2 + b*x)*sqrt(x))/(b*x^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(3/2)/x**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.30394, size = 74, normalized size = 0.89 \begin{align*} \frac{1}{4} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b}{c^{2} x^{2}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^(9/2),x, algorithm="giac")

[Out]

1/4*c^2*(3*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) - (5*(c*x + b)^(3/2) - 3*sqrt(c*x + b)*b)/(c^2*x^2))