\(\int \frac {(c x^2)^{3/2} (a+b x)^n}{x^6} \, dx\) [937]

Optimal result

Integrand size = 20, antiderivative size = 51 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=-\frac {b^2 c \sqrt {c x^2} (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{a^3 (1+n) x} \]

[Out]

-b^2*c*(b*x+a)^(1+n)*hypergeom([3, 1+n],[2+n],1+b*x/a)*(c*x^2)^(1/2)/a^3/(1+n)/x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 67} \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=-\frac {b^2 c \sqrt {c x^2} (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (3,n+1,n+2,\frac {b x}{a}+1\right )}{a^3 (n+1) x} \]

[In]

Int[((c*x^2)^(3/2)*(a + b*x)^n)/x^6,x]

[Out]

-((b^2*c*Sqrt[c*x^2]*(a + b*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (b*x)/a])/(a^3*(1 + n)*x))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {(a+b x)^n}{x^3} \, dx}{x} \\ & = -\frac {b^2 c \sqrt {c x^2} (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.98 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=-\frac {b^2 \left (c x^2\right )^{3/2} (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{a^3 (1+n) x^3} \]

[In]

Integrate[((c*x^2)^(3/2)*(a + b*x)^n)/x^6,x]

[Out]

-((b^2*(c*x^2)^(3/2)*(a + b*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (b*x)/a])/(a^3*(1 + n)*x^3))

Maple [F]

\[\int \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right )^{n}}{x^{6}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=\int { \frac {\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{n}}{x^{6}} \,d x } \]

[In]

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

[Out]

integral(sqrt(c*x^2)*(b*x + a)^n*c/x^4, x)

Sympy [F]

\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{n}}{x^{6}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=\int { \frac {\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{n}}{x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=\int { \frac {\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{n}}{x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^6} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^n}{x^6} \,d x \]

[In]

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

[Out]

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