Integrand size = 20, antiderivative size = 50 \[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\frac {b^3 x (a+b x)^{1+p} \operatorname {Hypergeometric2F1}\left (4,1+p,2+p,1+\frac {b x}{a}\right )}{a^4 c (1+p) \sqrt {c x^2}} \] Output:
b^3*x*(b*x+a)^(p+1)*hypergeom([4, p+1],[2+p],1+b*x/a)/a^4/c/(p+1)/(c*x^2)^ (1/2)
Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\frac {b^3 c x^5 (a+b x)^{1+p} \operatorname {Hypergeometric2F1}\left (4,1+p,2+p,1+\frac {b x}{a}\right )}{a^4 (1+p) \left (c x^2\right )^{5/2}} \] Input:
Integrate[(a + b*x)^p/(x*(c*x^2)^(3/2)),x]
Output:
(b^3*c*x^5*(a + b*x)^(1 + p)*Hypergeometric2F1[4, 1 + p, 2 + p, 1 + (b*x)/ a])/(a^4*(1 + p)*(c*x^2)^(5/2))
Time = 0.26 (sec) , antiderivative size = 50, 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 = {30, 75}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle \frac {x \int \frac {(a+b x)^p}{x^4}dx}{c \sqrt {c x^2}}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {b^3 x (a+b x)^{p+1} \operatorname {Hypergeometric2F1}\left (4,p+1,p+2,\frac {b x}{a}+1\right )}{a^4 c (p+1) \sqrt {c x^2}}\) |
Input:
Int[(a + b*x)^p/(x*(c*x^2)^(3/2)),x]
Output:
(b^3*x*(a + b*x)^(1 + p)*Hypergeometric2F1[4, 1 + p, 2 + p, 1 + (b*x)/a])/ (a^4*c*(1 + p)*Sqrt[c*x^2])
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
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] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
\[\int \frac {\left (b x +a \right )^{p}}{x \left (c \,x^{2}\right )^{\frac {3}{2}}}d x\]
Input:
int((b*x+a)^p/x/(c*x^2)^(3/2),x)
Output:
int((b*x+a)^p/x/(c*x^2)^(3/2),x)
\[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{p}}{\left (c x^{2}\right )^{\frac {3}{2}} x} \,d x } \] Input:
integrate((b*x+a)^p/x/(c*x^2)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(c*x^2)*(b*x + a)^p/(c^2*x^5), x)
\[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{p}}{x \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x+a)**p/x/(c*x**2)**(3/2),x)
Output:
Integral((a + b*x)**p/(x*(c*x**2)**(3/2)), x)
\[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{p}}{\left (c x^{2}\right )^{\frac {3}{2}} x} \,d x } \] Input:
integrate((b*x+a)^p/x/(c*x^2)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x + a)^p/((c*x^2)^(3/2)*x), x)
\[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{p}}{\left (c x^{2}\right )^{\frac {3}{2}} x} \,d x } \] Input:
integrate((b*x+a)^p/x/(c*x^2)^(3/2),x, algorithm="giac")
Output:
integrate((b*x + a)^p/((c*x^2)^(3/2)*x), x)
Timed out. \[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^p}{x\,{\left (c\,x^2\right )}^{3/2}} \,d x \] Input:
int((a + b*x)^p/(x*(c*x^2)^(3/2)),x)
Output:
int((a + b*x)^p/(x*(c*x^2)^(3/2)), x)
\[ \int \frac {(a+b x)^p}{x \left (c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \left (b x +a \right )^{p} a^{2}-\left (b x +a \right )^{p} a b p x -\left (b x +a \right )^{p} b^{2} p^{2} x^{2}+2 \left (b x +a \right )^{p} b^{2} p \,x^{2}+\left (\int \frac {\left (b x +a \right )^{p}}{b \,x^{2}+a x}d x \right ) b^{3} p^{3} x^{3}-3 \left (\int \frac {\left (b x +a \right )^{p}}{b \,x^{2}+a x}d x \right ) b^{3} p^{2} x^{3}+2 \left (\int \frac {\left (b x +a \right )^{p}}{b \,x^{2}+a x}d x \right ) b^{3} p \,x^{3}\right )}{6 a^{2} c^{2} x^{3}} \] Input:
int((b*x+a)^p/x/(c*x^2)^(3/2),x)
Output:
(sqrt(c)*( - 2*(a + b*x)**p*a**2 - (a + b*x)**p*a*b*p*x - (a + b*x)**p*b** 2*p**2*x**2 + 2*(a + b*x)**p*b**2*p*x**2 + int((a + b*x)**p/(a*x + b*x**2) ,x)*b**3*p**3*x**3 - 3*int((a + b*x)**p/(a*x + b*x**2),x)*b**3*p**2*x**3 + 2*int((a + b*x)**p/(a*x + b*x**2),x)*b**3*p*x**3))/(6*a**2*c**2*x**3)