Integrand size = 22, antiderivative size = 62 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\frac {c (d x)^{3+m} \sqrt {c x^2} (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (4+m,-p,5+m,-\frac {b x}{a}\right )}{d^3 (4+m)} \] Output:
c*(d*x)^(3+m)*(c*x^2)^(1/2)*(b*x+a)^p*hypergeom([-p, 4+m],[5+m],-b*x/a)/d^ 3/(4+m)/((1+b*x/a)^p)
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\frac {x (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (4+m,-p,5+m,-\frac {b x}{a}\right )}{4+m} \] Input:
Integrate[(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^p,x]
Output:
(x*(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^p*Hypergeometric2F1[4 + m, -p, 5 + m, - ((b*x)/a)])/((4 + m)*(1 + (b*x)/a)^p)
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {30, 76, 74}
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.
\(\displaystyle \int \left (c x^2\right )^{3/2} (d x)^m (a+b x)^p \, dx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle \frac {c \sqrt {c x^2} \int (d x)^{m+3} (a+b x)^pdx}{d^3 x}\) |
\(\Big \downarrow \) 76 |
\(\displaystyle \frac {c \sqrt {c x^2} (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \int (d x)^{m+3} \left (\frac {b x}{a}+1\right )^pdx}{d^3 x}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {c \sqrt {c x^2} (d x)^{m+4} (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (m+4,-p,m+5,-\frac {b x}{a}\right )}{d^4 (m+4) x}\) |
Input:
Int[(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^p,x]
Output:
(c*(d*x)^(4 + m)*Sqrt[c*x^2]*(a + b*x)^p*Hypergeometric2F1[4 + m, -p, 5 + m, -((b*x)/a)])/(d^4*(4 + m)*x*(1 + (b*x)/a)^p)
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^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart [n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d* (x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !Integer Q[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0] && ((RationalQ[m] && !(EqQ[n, -2 ^(-1)] && EqQ[c^2 - d^2, 0])) || !RationalQ[n])
\[\int \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right )^{p}d x\]
Input:
int((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^p,x)
Output:
int((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^p,x)
\[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\int { \left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{p} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^p,x, algorithm="fricas")
Output:
integral(sqrt(c*x^2)*(b*x + a)^p*(d*x)^m*c*x^2, x)
\[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\int \left (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m} \left (a + b x\right )^{p}\, dx \] Input:
integrate((d*x)**m*(c*x**2)**(3/2)*(b*x+a)**p,x)
Output:
Integral((c*x**2)**(3/2)*(d*x)**m*(a + b*x)**p, x)
\[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\int { \left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{p} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^p,x, algorithm="maxima")
Output:
integrate((c*x^2)^(3/2)*(b*x + a)^p*(d*x)^m, x)
\[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\int { \left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{p} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^p,x, algorithm="giac")
Output:
integrate((c*x^2)^(3/2)*(b*x + a)^p*(d*x)^m, x)
Timed out. \[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\int {\left (d\,x\right )}^m\,{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^p \,d x \] Input:
int((d*x)^m*(c*x^2)^(3/2)*(a + b*x)^p,x)
Output:
int((d*x)^m*(c*x^2)^(3/2)*(a + b*x)^p, x)
\[ \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^p \, dx=\text {too large to display} \] Input:
int((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^p,x)
Output:
(d**m*sqrt(c)*c*( - x**m*(a + b*x)**p*a**4*m**3*p - 6*x**m*(a + b*x)**p*a* *4*m**2*p - 11*x**m*(a + b*x)**p*a**4*m*p - 6*x**m*(a + b*x)**p*a**4*p + x **m*(a + b*x)**p*a**3*b*m**3*p*x + x**m*(a + b*x)**p*a**3*b*m**2*p**2*x + 5*x**m*(a + b*x)**p*a**3*b*m**2*p*x + 5*x**m*(a + b*x)**p*a**3*b*m*p**2*x + 6*x**m*(a + b*x)**p*a**3*b*m*p*x + 6*x**m*(a + b*x)**p*a**3*b*p**2*x - x **m*(a + b*x)**p*a**2*b**2*m**3*p*x**2 - 2*x**m*(a + b*x)**p*a**2*b**2*m** 2*p**2*x**2 - 4*x**m*(a + b*x)**p*a**2*b**2*m**2*p*x**2 - x**m*(a + b*x)** p*a**2*b**2*m*p**3*x**2 - 7*x**m*(a + b*x)**p*a**2*b**2*m*p**2*x**2 - 3*x* *m*(a + b*x)**p*a**2*b**2*m*p*x**2 - 3*x**m*(a + b*x)**p*a**2*b**2*p**3*x* *2 - 3*x**m*(a + b*x)**p*a**2*b**2*p**2*x**2 + x**m*(a + b*x)**p*a*b**3*m* *3*p*x**3 + 3*x**m*(a + b*x)**p*a*b**3*m**2*p**2*x**3 + 3*x**m*(a + b*x)** p*a*b**3*m**2*p*x**3 + 3*x**m*(a + b*x)**p*a*b**3*m*p**3*x**3 + 6*x**m*(a + b*x)**p*a*b**3*m*p**2*x**3 + 2*x**m*(a + b*x)**p*a*b**3*m*p*x**3 + x**m* (a + b*x)**p*a*b**3*p**4*x**3 + 3*x**m*(a + b*x)**p*a*b**3*p**3*x**3 + 2*x **m*(a + b*x)**p*a*b**3*p**2*x**3 + x**m*(a + b*x)**p*b**4*m**4*x**4 + 4*x **m*(a + b*x)**p*b**4*m**3*p*x**4 + 6*x**m*(a + b*x)**p*b**4*m**3*x**4 + 6 *x**m*(a + b*x)**p*b**4*m**2*p**2*x**4 + 18*x**m*(a + b*x)**p*b**4*m**2*p* x**4 + 11*x**m*(a + b*x)**p*b**4*m**2*x**4 + 4*x**m*(a + b*x)**p*b**4*m*p* *3*x**4 + 18*x**m*(a + b*x)**p*b**4*m*p**2*x**4 + 22*x**m*(a + b*x)**p*b** 4*m*p*x**4 + 6*x**m*(a + b*x)**p*b**4*m*x**4 + x**m*(a + b*x)**p*b**4*p...