Integrand size = 22, antiderivative size = 468 \[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=-\frac {2 c (11+8 p) \sqrt {c+d x} \left (a+b x^2\right )^{1+p}}{b d^2 (5+4 p) (7+4 p)}+\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{1+p}}{b d^2 (7+4 p)}+\frac {2 c \left (a d^2 (11+8 p)-8 b c^2 \left (3+5 p+2 p^2\right )\right ) \sqrt {c+d x} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{b d^4 (5+4 p) (7+4 p)}-\frac {2 \left (3 a d-\frac {8 b c^2 \left (3+5 p+2 p^2\right )}{d (5+4 p)}\right ) (c+d x)^{3/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 b d^3 (7+4 p)} \] Output:
-2*c*(11+8*p)*(d*x+c)^(1/2)*(b*x^2+a)^(p+1)/b/d^2/(5+4*p)/(7+4*p)+2*(d*x+c )^(3/2)*(b*x^2+a)^(p+1)/b/d^2/(7+4*p)+2*c*(a*d^2*(11+8*p)-8*b*c^2*(2*p^2+5 *p+3))*(d*x+c)^(1/2)*(b*x^2+a)^p*AppellF1(1/2,-p,-p,3/2,(d*x+c)/(c-(-a)^(1 /2)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^4/(5+4*p)/(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)-2/3*(3*a*d-8*b*c^2*(2*p^2+5*p+3)/d/(5+4*p))*(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)
Time = 0.43 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.34 \[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\frac {2 (c+d x)^{7/2} \left (a+b x^2\right )^p \left (\frac {a d^2-b c d x+\sqrt {-a b d^2} (c+d x)}{\left (-b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )^{-p} \left (\frac {-a d^2+b c d x+\sqrt {-a b d^2} (c+d x)}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )^{-p} \left (\left (15+92 p+144 p^2+64 p^3\right ) \operatorname {AppellF1}\left (-\frac {7}{2}-2 p,-p,-p,-\frac {5}{2}-2 p,\frac {b c^2+a d^2}{\left (b c-\sqrt {-a b d^2}\right ) (c+d x)},\frac {b c^2+a d^2}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )-\frac {c (7+4 p) \left (\left (9+48 p+48 p^2\right ) \operatorname {AppellF1}\left (-\frac {5}{2}-2 p,-p,-p,-\frac {3}{2}-2 p,\frac {b c^2+a d^2}{\left (b c-\sqrt {-a b d^2}\right ) (c+d x)},\frac {b c^2+a d^2}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )+\frac {c (5+4 p) \left (-3 (1+4 p) \operatorname {AppellF1}\left (-\frac {3}{2}-2 p,-p,-p,-\frac {1}{2}-2 p,\frac {b c^2+a d^2}{\left (b c-\sqrt {-a b d^2}\right ) (c+d x)},\frac {b c^2+a d^2}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )+\frac {c (3+4 p) \operatorname {AppellF1}\left (-\frac {1}{2}-2 p,-p,-p,\frac {1}{2}-2 p,\frac {b c^2+a d^2}{\left (b c-\sqrt {-a b d^2}\right ) (c+d x)},\frac {b c^2+a d^2}{\left (b c+\sqrt {-a b d^2}\right ) (c+d x)}\right )}{c+d x}\right )}{c+d x}\right )}{c+d x}\right )}{d^4 (1+4 p) (3+4 p) (5+4 p) (7+4 p)} \] Input:
Integrate[(x^3*(a + b*x^2)^p)/Sqrt[c + d*x],x]
Output:
(2*(c + d*x)^(7/2)*(a + b*x^2)^p*((15 + 92*p + 144*p^2 + 64*p^3)*AppellF1[ -7/2 - 2*p, -p, -p, -5/2 - 2*p, (b*c^2 + a*d^2)/((b*c - Sqrt[-(a*b*d^2)])* (c + d*x)), (b*c^2 + a*d^2)/((b*c + Sqrt[-(a*b*d^2)])*(c + d*x))] - (c*(7 + 4*p)*((9 + 48*p + 48*p^2)*AppellF1[-5/2 - 2*p, -p, -p, -3/2 - 2*p, (b*c^ 2 + a*d^2)/((b*c - Sqrt[-(a*b*d^2)])*(c + d*x)), (b*c^2 + a*d^2)/((b*c + S qrt[-(a*b*d^2)])*(c + d*x))] + (c*(5 + 4*p)*(-3*(1 + 4*p)*AppellF1[-3/2 - 2*p, -p, -p, -1/2 - 2*p, (b*c^2 + a*d^2)/((b*c - Sqrt[-(a*b*d^2)])*(c + d* x)), (b*c^2 + a*d^2)/((b*c + Sqrt[-(a*b*d^2)])*(c + d*x))] + (c*(3 + 4*p)* AppellF1[-1/2 - 2*p, -p, -p, 1/2 - 2*p, (b*c^2 + a*d^2)/((b*c - Sqrt[-(a*b *d^2)])*(c + d*x)), (b*c^2 + a*d^2)/((b*c + Sqrt[-(a*b*d^2)])*(c + d*x))]) /(c + d*x)))/(c + d*x)))/(c + d*x)))/(d^4*(1 + 4*p)*(3 + 4*p)*(5 + 4*p)*(7 + 4*p)*((a*d^2 - b*c*d*x + Sqrt[-(a*b*d^2)]*(c + d*x))/((-(b*c) + Sqrt[-( a*b*d^2)])*(c + d*x)))^p*((-(a*d^2) + b*c*d*x + Sqrt[-(a*b*d^2)]*(c + d*x) )/((b*c + Sqrt[-(a*b*d^2)])*(c + d*x)))^p)
Leaf count is larger than twice the leaf count of optimal. \(1406\) vs. \(2(468)=936\).
Time = 2.26 (sec) , antiderivative size = 1406, normalized size of antiderivative = 3.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {624, 624, 596, 624, 514, 150, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 624 |
| \(\displaystyle \frac {\int x^2 \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {c \int \frac {x^2 \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}\) |
| \(\Big \downarrow \) 624 |
| \(\displaystyle \frac {\frac {\int x (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int x \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {c \int \frac {x \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 596 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}}{d}-\frac {c \int \frac {x \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 624 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}}{d}-\frac {c \left (\frac {\int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {c \int \frac {\left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 514 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}}{d}-\frac {c \left (\frac {\left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int \sqrt {c+d x} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^pd(c+d x)}{d^2}-\frac {c \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \int \frac {\left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^p}{\sqrt {c+d x}}d(c+d x)}{d^2}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 d^2}-\frac {2 c \sqrt {c+d x} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (a d^2+b c^2\right ) \int \frac {\left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}-\frac {b c \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}}{b (4 p+5)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (a d^2+b c^2\right ) \int \frac {\left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}-\frac {b c \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}}{b (4 p+5)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 d^2}-\frac {2 c \sqrt {c+d x} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 514 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (b x^2+a\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (b c^2+a d^2\right ) \left (b x^2+a\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} \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 (b x^2+a\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} \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 (b x^2+a\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (b c^2+a d^2\right ) \left (b x^2+a\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} \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 (b x^2+a\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} \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}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (b x^2+a\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (b c^2+a d^2\right ) \left (b x^2+a\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} \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 (b x^2+a\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} \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)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (b x^2+a\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 d^2}-\frac {2 c \sqrt {c+d x} \left (b x^2+a\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 {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}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {\frac {\frac {2 (c+d x)^{3/2} \left (b x^2+a\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {2 \left (b c^2+a d^2\right ) (c+d x)^{3/2} \left (b x^2+a\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 d^2}-\frac {2 b c (c+d x)^{5/2} \left (b x^2+a\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^2}\right )}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (b x^2+a\right )^{p+1}}{b (4 p+5)}-\frac {\frac {2 \left (b c^2+a d^2\right ) \sqrt {c+d x} \left (b x^2+a\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 {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 (b x^2+a\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 d^2}}{b (4 p+5)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 \sqrt {c+d x} \left (b x^2+a\right )^{p+1}}{b (4 p+5)}-\frac {\frac {2 \left (b c^2+a d^2\right ) \sqrt {c+d x} \left (b x^2+a\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 {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 (b x^2+a\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 d^2}}{b (4 p+5)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (b x^2+a\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 d^2}-\frac {2 c \sqrt {c+d x} \left (b x^2+a\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 {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}\right )}{d}\right )}{d}\) |
Input:
Int[(x^3*(a + b*x^2)^p)/Sqrt[c + d*x],x]
Output:
-((c*(-((c*((-2*c*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 + (S qrt[-a]*d)/Sqrt[b]))^p) + (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 - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)))/d) + ((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)/Sqrt[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)/S qrt[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))/d) + (-((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)/Sqrt[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 + ...
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]
\[\int \frac {x^{3} \left (b \,x^{2}+a \right )^{p}}{\sqrt {d x +c}}d x\]
Input:
int(x^3*(b*x^2+a)^p/(d*x+c)^(1/2),x)
Output:
int(x^3*(b*x^2+a)^p/(d*x+c)^(1/2),x)
\[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(b*x^2+a)^p/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^p*x^3/sqrt(d*x + c), x)
\[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\int \frac {x^{3} \left (a + b x^{2}\right )^{p}}{\sqrt {c + d x}}\, dx \] Input:
integrate(x**3*(b*x**2+a)**p/(d*x+c)**(1/2),x)
Output:
Integral(x**3*(a + b*x**2)**p/sqrt(c + d*x), x)
\[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(b*x^2+a)^p/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^p*x^3/sqrt(d*x + c), x)
\[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x^3*(b*x^2+a)^p/(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^p*x^3/sqrt(d*x + c), x)
Timed out. \[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\int \frac {x^3\,{\left (b\,x^2+a\right )}^p}{\sqrt {c+d\,x}} \,d x \] Input:
int((x^3*(a + b*x^2)^p)/(c + d*x)^(1/2),x)
Output:
int((x^3*(a + b*x^2)^p)/(c + d*x)^(1/2), x)
\[ \int \frac {x^3 \left (a+b x^2\right )^p}{\sqrt {c+d x}} \, dx=\text {too large to display} \] Input:
int(x^3*(b*x^2+a)^p/(d*x+c)^(1/2),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 + 8*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c**2*d*p + 12*sq rt(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 + 32*s qrt(c + d*x)*(a + b*x**2)**p*b**2*c**3*p**2*x + 80*sqrt(c + d*x)*(a + b*x* *2)**p*b**2*c**3*p*x + 48*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**3*x - 32*s qrt(c + d*x)*(a + b*x**2)**p*b**2*c**2*d*p**2*x**2 - 72*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**2*d*p*x**2 - 36*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*sqr t(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)/(6 4*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((sqrt(c + d*x)*(a + b*x**2)**p*x**2)/(64*a*c*p**3 + 240*a*c*...