Integrand size = 22, antiderivative size = 373 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{1+p}}{b d (7+4 p)}-\frac {2 \left (3 a d^2-4 b c^2 (1+p)\right ) (c+d x)^{3/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 b d^3 (7+4 p)}-\frac {8 c (1+p) (c+d x)^{5/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{5 d^3 (7+4 p)} \] Output:
2*(d*x+c)^(3/2)*(b*x^2+a)^(p+1)/b/d/(7+4*p)-2/3*(3*a*d^2-4*b*c^2*(p+1))*(d *x+c)^(3/2)*(b*x^2+a)^p*AppellF1(3/2,-p,-p,5/2,(d*x+c)/(c-(-a)^(1/2)*d/b^( 1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^3/(7+4*p)/((1-(d*x+c)/(c-(-a)^ (1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)-8/5*c*(p+1)* (d*x+c)^(5/2)*(b*x^2+a)^p*AppellF1(5/2,-p,-p,7/2,(d*x+c)/(c-(-a)^(1/2)*d/b ^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^3/(7+4*p)/((1-(d*x+c)/(c-(-a)^ (1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)
Time = 2.55 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.87 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\frac {2 \left (\frac {d \left (\sqrt {-\frac {a b}{d^2}} d-b x\right )}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )^{-p} (c+d x)^{3/2} \left (\frac {a-\sqrt {-\frac {a b}{d^2}} d x}{a+c \sqrt {-\frac {a b}{d^2}}}\right )^{-p} \left (a+b x^2\right )^p \left (35 c^2 \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )-3 (c+d x) \left (14 c \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )-5 (c+d x) \operatorname {AppellF1}\left (\frac {7}{2},-p,-p,\frac {9}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )\right )\right )}{105 d^3} \] Input:
Integrate[x^2*Sqrt[c + d*x]*(a + b*x^2)^p,x]
Output:
(2*(c + d*x)^(3/2)*(a + b*x^2)^p*(35*c^2*AppellF1[3/2, -p, -p, 5/2, (c + d *x)/(c + a/Sqrt[-((a*b)/d^2)]), (b*(c + d*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^ 2)] - 3*(c + d*x)*(14*c*AppellF1[5/2, -p, -p, 7/2, (c + d*x)/(c + a/Sqrt[- ((a*b)/d^2)]), (b*(c + d*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^2)] - 5*(c + d*x) *AppellF1[7/2, -p, -p, 9/2, (c + d*x)/(c + a/Sqrt[-((a*b)/d^2)]), (b*(c + d*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^2)])))/(105*d^3*((d*(Sqrt[-((a*b)/d^2)]* d - b*x))/(b*c + Sqrt[-((a*b)/d^2)]*d^2))^p*((a - Sqrt[-((a*b)/d^2)]*d*x)/ (a + c*Sqrt[-((a*b)/d^2)]))^p)
Time = 1.19 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {624, 596, 719, 514, 150}
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.
\(\displaystyle \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx\) |
\(\Big \downarrow \) 624 |
\(\displaystyle \frac {\int x (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}\) |
\(\Big \downarrow \) 596 |
\(\displaystyle \frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}\right )}{d}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (a d^2+b c^2\right ) \int \frac {\left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}-\frac {b c \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}}{b (4 p+5)}\right )}{d}\) |
\(\Big \downarrow \) 514 |
\(\displaystyle \frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int \sqrt {c+d x} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}-\frac {b c \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int (c+d x)^{3/2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}\right )}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int \frac {\left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p}{\sqrt {c+d x}}d(c+d x)}{d^2}-\frac {b c \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int \sqrt {c+d x} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}}{b (4 p+5)}\right )}{d}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {2 (c+d x)^{3/2} \left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 d^2}-\frac {2 b c (c+d x)^{5/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{5 d^2}\right )}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\frac {2 \sqrt {c+d x} \left (a d^2+b c^2\right ) \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2}-\frac {2 b c (c+d x)^{3/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 d^2}}{b (4 p+5)}\right )}{d}\) |
Input:
Int[x^2*Sqrt[c + d*x]*(a + b*x^2)^p,x]
Output:
-((c*((2*Sqrt[c + d*x]*(a + b*x^2)^(1 + p))/(b*(5 + 4*p)) - ((2*(b*c^2 + a *d^2)*Sqrt[c + d*x]*(a + b*x^2)^p*AppellF1[1/2, -p, -p, 3/2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/S qrt[b]))^p) - (2*b*c*(c + d*x)^(3/2)*(a + b*x^2)^p*AppellF1[3/2, -p, -p, 5 /2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt [b])])/(3*d^2*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/ (c + (Sqrt[-a]*d)/Sqrt[b]))^p))/(b*(5 + 4*p))))/d) + ((2*(c + d*x)^(3/2)*( a + b*x^2)^(1 + p))/(b*(7 + 4*p)) - (3*((2*(b*c^2 + a*d^2)*(c + d*x)^(3/2) *(a + b*x^2)^p*AppellF1[3/2, -p, -p, 5/2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt [b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(3*d^2*(1 - (c + d*x)/(c - (S qrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p) - (2* b*c*(c + d*x)^(5/2)*(a + b*x^2)^p*AppellF1[5/2, -p, -p, 7/2, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(5*d^2*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d) /Sqrt[b]))^p)))/(b*(7 + 4*p)))/d
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( c + d*x)/(c + d*q))^p) Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] - Simp[n/(b*(n + 2*p + 2)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && GtQ[n, 0] && NeQ[n + 2*p + 2, 0]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[1/d Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si mp[c/d Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
\[\int x^{2} \sqrt {d x +c}\, \left (b \,x^{2}+a \right )^{p}d x\]
Input:
int(x^2*(d*x+c)^(1/2)*(b*x^2+a)^p,x)
Output:
int(x^2*(d*x+c)^(1/2)*(b*x^2+a)^p,x)
\[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int { \sqrt {d x + c} {\left (b x^{2} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(1/2)*(b*x^2+a)^p,x, algorithm="fricas")
Output:
integral(sqrt(d*x + c)*(b*x^2 + a)^p*x^2, x)
\[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int x^{2} \left (a + b x^{2}\right )^{p} \sqrt {c + d x}\, dx \] Input:
integrate(x**2*(d*x+c)**(1/2)*(b*x**2+a)**p,x)
Output:
Integral(x**2*(a + b*x**2)**p*sqrt(c + d*x), x)
\[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int { \sqrt {d x + c} {\left (b x^{2} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(1/2)*(b*x^2+a)^p,x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)*(b*x^2 + a)^p*x^2, x)
\[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int { \sqrt {d x + c} {\left (b x^{2} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(1/2)*(b*x^2+a)^p,x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)*(b*x^2 + a)^p*x^2, x)
Timed out. \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int x^2\,{\left (b\,x^2+a\right )}^p\,\sqrt {c+d\,x} \,d x \] Input:
int(x^2*(a + b*x^2)^p*(c + d*x)^(1/2),x)
Output:
int(x^2*(a + b*x^2)^p*(c + d*x)^(1/2), x)
\[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:
int(x^2*(d*x+c)^(1/2)*(b*x^2+a)^p,x)
Output:
( - 24*sqrt(c + d*x)*(a + b*x**2)**p*a**2*d**3*p - 30*sqrt(c + d*x)*(a + b *x**2)**p*a**2*d**3 - 2*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c**2*d + 32*sqrt (c + d*x)*(a + b*x**2)**p*a*b*c*d**2*p**2*x + 40*sqrt(c + d*x)*(a + b*x**2 )**p*a*b*c*d**2*p*x - 8*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**3*p*x - 8*sq rt(c + d*x)*(a + b*x**2)**p*b**2*c**3*x + 8*sqrt(c + d*x)*(a + b*x**2)**p* b**2*c**2*d*p*x**2 + 6*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**2*d*x**2 + 32 *sqrt(c + d*x)*(a + b*x**2)**p*b**2*c*d**2*p**2*x**3 + 64*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c*d**2*p*x**3 + 30*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c *d**2*x**3 + 3072*int((sqrt(c + d*x)*(a + b*x**2)**p*x**2)/(64*a*c*p**3 + 240*a*c*p**2 + 284*a*c*p + 105*a*c + 64*a*d*p**3*x + 240*a*d*p**2*x + 284* a*d*p*x + 105*a*d*x + 64*b*c*p**3*x**2 + 240*b*c*p**2*x**2 + 284*b*c*p*x** 2 + 105*b*c*x**2 + 64*b*d*p**3*x**3 + 240*b*d*p**2*x**3 + 284*b*d*p*x**3 + 105*b*d*x**3),x)*a**2*b*d**4*p**5 + 16128*int((sqrt(c + d*x)*(a + b*x**2) **p*x**2)/(64*a*c*p**3 + 240*a*c*p**2 + 284*a*c*p + 105*a*c + 64*a*d*p**3* x + 240*a*d*p**2*x + 284*a*d*p*x + 105*a*d*x + 64*b*c*p**3*x**2 + 240*b*c* p**2*x**2 + 284*b*c*p*x**2 + 105*b*c*x**2 + 64*b*d*p**3*x**3 + 240*b*d*p** 2*x**3 + 284*b*d*p*x**3 + 105*b*d*x**3),x)*a**2*b*d**4*p**4 + 31872*int((s qrt(c + d*x)*(a + b*x**2)**p*x**2)/(64*a*c*p**3 + 240*a*c*p**2 + 284*a*c*p + 105*a*c + 64*a*d*p**3*x + 240*a*d*p**2*x + 284*a*d*p*x + 105*a*d*x + 64 *b*c*p**3*x**2 + 240*b*c*p**2*x**2 + 284*b*c*p*x**2 + 105*b*c*x**2 + 64...