\(\int x^3 \sqrt {c+d x} (a+b x^2)^p \, dx\) [76]

Optimal result

Integrand size = 22, antiderivative size = 470 \[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=-\frac {2 c (13+8 p) (c+d x)^{3/2} \left (a+b x^2\right )^{1+p}}{b d^2 (7+4 p) (9+4 p)}+\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{1+p}}{b d^2 (9+4 p)}+\frac {2 c \left (3 a d^2 (13+8 p)-8 b c^2 \left (3+5 p+2 p^2\right )\right ) (c+d x)^{3/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{3 b d^4 (7+4 p) (9+4 p)}-\frac {2 \left (5 a d^2-\frac {8 b c^2 \left (3+5 p+2 p^2\right )}{7+4 p}\right ) (c+d x)^{5/2} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{5 b d^4 (9+4 p)} \] Output:

-2*c*(13+8*p)*(d*x+c)^(3/2)*(b*x^2+a)^(p+1)/b/d^2/(7+4*p)/(9+4*p)+2*(d*x+c 
)^(5/2)*(b*x^2+a)^(p+1)/b/d^2/(9+4*p)+2/3*c*(3*a*d^2*(13+8*p)-8*b*c^2*(2*p 
^2+5*p+3))*(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^4/(7+4*p)/(9+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)-2/5*(5*a*d^2-8*b*c^2*(2*p^2+5*p+3)/(7+4*p))*(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)))/b/d^4/(9+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)
 

Mathematica [A] (warning: unable to verify)

Time = 3.33 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.85 \[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\frac {2 \left (\frac {d \left (\sqrt {-\frac {a b}{d^2}} d-b x\right )}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )^{-p} (c+d x)^{3/2} \left (\frac {a-\sqrt {-\frac {a b}{d^2}} d x}{a+c \sqrt {-\frac {a b}{d^2}}}\right )^{-p} \left (a+b x^2\right )^p \left (-105 c^3 \operatorname {AppellF1}\left (\frac {3}{2},-p,-p,\frac {5}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )+189 c^2 (c+d x) \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )-135 c (c+d x)^2 \operatorname {AppellF1}\left (\frac {7}{2},-p,-p,\frac {9}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )+35 (c+d x)^3 \operatorname {AppellF1}\left (\frac {9}{2},-p,-p,\frac {11}{2},\frac {c+d x}{c+\frac {a}{\sqrt {-\frac {a b}{d^2}}}},\frac {b (c+d x)}{b c+\sqrt {-\frac {a b}{d^2}} d^2}\right )\right )}{315 d^4} \] Input:

Integrate[x^3*Sqrt[c + d*x]*(a + b*x^2)^p,x]
 

Output:

(2*(c + d*x)^(3/2)*(a + b*x^2)^p*(-105*c^3*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)] + 189*c^2*(c + d*x)*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)] - 135*c*(c 
+ d*x)^2*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)] + 35*(c + d*x)^3*AppellF1[9/ 
2, -p, -p, 11/2, (c + d*x)/(c + a/Sqrt[-((a*b)/d^2)]), (b*(c + d*x))/(b*c 
+ Sqrt[-((a*b)/d^2)]*d^2)]))/(315*d^4*((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)
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1467\) vs. \(2(470)=940\).

Time = 2.35 (sec) , antiderivative size = 1467, normalized size of antiderivative = 3.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {624, 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.

\(\displaystyle \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 624

\(\displaystyle \frac {\int x^2 (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x^2 \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}\)

\(\Big \downarrow \) 624

\(\displaystyle \frac {\frac {\int x (c+d x)^{5/2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int x (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}\right )}{d}\)

\(\Big \downarrow \) 596

\(\displaystyle \frac {\frac {\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{p+1}}{b (4 p+9)}-\frac {5 \int (a d-b c x) (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{b (4 p+9)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \int (a d-b c x) \sqrt {c+d x} \left (b x^2+a\right )^pdx}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{b (4 p+5)}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\frac {2 (c+d x)^{5/2} \left (a+b x^2\right )^{p+1}}{b (4 p+9)}-\frac {5 \left (\frac {\left (a d^2+b c^2\right ) \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{5/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+9)}}{d}-\frac {c \left (\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+7)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {2 (c+d x)^{3/2} \left (a+b x^2\right )^{p+1}}{b (4 p+7)}-\frac {3 \left (\frac {\left (a d^2+b c^2\right ) \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}-\frac {b c \int (c+d x)^{3/2} \left (b x^2+a\right )^pdx}{d}\right )}{b (4 p+7)}}{d}-\frac {c \left (\frac {2 \sqrt {c+d x} \left (a+b x^2\right )^{p+1}}{b (4 p+5)}-\frac {\frac {\left (a d^2+b c^2\right ) \int \frac {\left (b x^2+a\right )^p}{\sqrt {c+d x}}dx}{d}-\frac {b c \int \sqrt {c+d x} \left (b x^2+a\right )^pdx}{d}}{b (4 p+5)}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {\frac {\frac {2 (c+d x)^{5/2} \left (b x^2+a\right )^{p+1}}{b (4 p+9)}-\frac {5 \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 (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}-\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)^{5/2} \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+9)}}{d}-\frac {c \left (\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)}\right )}{d}}{d}-\frac {c \left (\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}\right )}{d}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {\frac {2 (c+d x)^{5/2} \left (b x^2+a\right )^{p+1}}{b (4 p+9)}-\frac {5 \left (\frac {2 \left (b c^2+a d^2\right ) (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}-\frac {2 b c (c+d x)^{7/2} \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 {7}{2},-p,-p,\frac {9}{2},\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{7 d^2}\right )}{b (4 p+9)}}{d}-\frac {c \left (\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)}\right )}{d}}{d}-\frac {c \left (\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}\right )}{d}\)

Input:

Int[x^3*Sqrt[c + d*x]*(a + b*x^2)^p,x]
 

Output:

-((c*(-((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 + 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 - (Sqrt[-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))/d) + (-((c*((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 - (Sq 
rt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p) - (...
 

Defintions of rubi rules used

rule 150

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])
 

rule 514

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]
 

rule 596

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]
 

rule 624

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]
 

rule 719

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]
 

Maple [F]

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

Input:

int(x^3*(d*x+c)^(1/2)*(b*x^2+a)^p,x)
 

Output:

int(x^3*(d*x+c)^(1/2)*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int { \sqrt {d x + c} {\left (b x^{2} + a\right )}^{p} x^{3} \,d x } \] Input:

integrate(x^3*(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^3, x)
 

Sympy [F]

\[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int x^{3} \left (a + b x^{2}\right )^{p} \sqrt {c + d x}\, dx \] Input:

integrate(x**3*(d*x+c)**(1/2)*(b*x**2+a)**p,x)
 

Output:

Integral(x**3*(a + b*x**2)**p*sqrt(c + d*x), x)
 

Maxima [F]

\[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int { \sqrt {d x + c} {\left (b x^{2} + a\right )}^{p} x^{3} \,d x } \] Input:

integrate(x^3*(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^3, x)
 

Giac [F]

\[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\int { \sqrt {d x + c} {\left (b x^{2} + a\right )}^{p} x^{3} \,d x } \] Input:

integrate(x^3*(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^3, x)
 

Mupad [F(-1)]

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

Reduce [F]

\[ \int x^3 \sqrt {c+d x} \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:

int(x^3*(d*x+c)^(1/2)*(b*x^2+a)^p,x)
 

Output:

(2*( - 64*sqrt(c + d*x)*(a + b*x**2)**p*a**2*d**3*p**2 - 184*sqrt(c + d*x) 
*(a + b*x**2)**p*a**2*d**3*p - 120*sqrt(c + d*x)*(a + b*x**2)**p*a**2*d**3 
 + 4*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c**2*d*p + 6*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 + 
48*sqrt(c + d*x)*(a + b*x**2)**p*a*b*c*d**2*p*x + 64*sqrt(c + d*x)*(a + b* 
x**2)**p*a*b*d**3*p**3*x**2 + 160*sqrt(c + d*x)*(a + b*x**2)**p*a*b*d**3*p 
**2*x**2 + 84*sqrt(c + d*x)*(a + b*x**2)**p*a*b*d**3*p*x**2 + 16*sqrt(c + 
d*x)*(a + b*x**2)**p*b**2*c**3*p**2*x + 40*sqrt(c + d*x)*(a + b*x**2)**p*b 
**2*c**3*p*x + 24*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**3*x - 16*sqrt(c + 
d*x)*(a + b*x**2)**p*b**2*c**2*d*p**2*x**2 - 36*sqrt(c + d*x)*(a + b*x**2) 
**p*b**2*c**2*d*p*x**2 - 18*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c**2*d*x**2 
 + 16*sqrt(c + d*x)*(a + b*x**2)**p*b**2*c*d**2*p**2*x**3 + 32*sqrt(c + d* 
x)*(a + b*x**2)**p*b**2*c*d**2*p*x**3 + 15*sqrt(c + d*x)*(a + b*x**2)**p*b 
**2*c*d**2*x**3 + 64*sqrt(c + d*x)*(a + b*x**2)**p*b**2*d**3*p**3*x**4 + 2 
40*sqrt(c + d*x)*(a + b*x**2)**p*b**2*d**3*p**2*x**4 + 284*sqrt(c + d*x)*( 
a + b*x**2)**p*b**2*d**3*p*x**4 + 105*sqrt(c + d*x)*(a + b*x**2)**p*b**2*d 
**3*x**4 - 8192*int((sqrt(c + d*x)*(a + b*x**2)**p*x**2)/(256*a*c*p**4 + 1 
536*a*c*p**3 + 3296*a*c*p**2 + 2976*a*c*p + 945*a*c + 256*a*d*p**4*x + 153 
6*a*d*p**3*x + 3296*a*d*p**2*x + 2976*a*d*p*x + 945*a*d*x + 256*b*c*p**4*x 
**2 + 1536*b*c*p**3*x**2 + 3296*b*c*p**2*x**2 + 2976*b*c*p*x**2 + 945*b...