Integrand size = 21, antiderivative size = 245 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=-\frac {(b c-a d)^3 x}{9 c d^3 \left (c+d x^2\right )^{9/2}}+\frac {(b c-a d)^2 (19 b c+8 a d) x}{63 c^2 d^3 \left (c+d x^2\right )^{7/2}}-\frac {(b c-a d) \left (25 b^2 c^2+22 a b c d+16 a^2 d^2\right ) x}{105 c^3 d^3 \left (c+d x^2\right )^{5/2}}+\frac {\left (5 b^3 c^3+12 a b^2 c^2 d+24 a^2 b c d^2+64 a^3 d^3\right ) x}{315 c^4 d^3 \left (c+d x^2\right )^{3/2}}+\frac {2 \left (5 b^3 c^3+12 a b^2 c^2 d+24 a^2 b c d^2+64 a^3 d^3\right ) x}{315 c^5 d^3 \sqrt {c+d x^2}} \] Output:
-1/9*(-a*d+b*c)^3*x/c/d^3/(d*x^2+c)^(9/2)+1/63*(-a*d+b*c)^2*(8*a*d+19*b*c) *x/c^2/d^3/(d*x^2+c)^(7/2)-1/105*(-a*d+b*c)*(16*a^2*d^2+22*a*b*c*d+25*b^2* c^2)*x/c^3/d^3/(d*x^2+c)^(5/2)+1/315*(64*a^3*d^3+24*a^2*b*c*d^2+12*a*b^2*c ^2*d+5*b^3*c^3)*x/c^4/d^3/(d*x^2+c)^(3/2)+2/315*(64*a^3*d^3+24*a^2*b*c*d^2 +12*a*b^2*c^2*d+5*b^3*c^3)*x/c^5/d^3/(d*x^2+c)^(1/2)
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=\frac {5 b^3 c^3 x^7 \left (9 c+2 d x^2\right )+3 a b^2 c^2 x^5 \left (63 c^2+36 c d x^2+8 d^2 x^4\right )+3 a^2 b c x^3 \left (105 c^3+126 c^2 d x^2+72 c d^2 x^4+16 d^3 x^6\right )+a^3 \left (315 c^4 x+840 c^3 d x^3+1008 c^2 d^2 x^5+576 c d^3 x^7+128 d^4 x^9\right )}{315 c^5 \left (c+d x^2\right )^{9/2}} \] Input:
Integrate[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]
Output:
(5*b^3*c^3*x^7*(9*c + 2*d*x^2) + 3*a*b^2*c^2*x^5*(63*c^2 + 36*c*d*x^2 + 8* d^2*x^4) + 3*a^2*b*c*x^3*(105*c^3 + 126*c^2*d*x^2 + 72*c*d^2*x^4 + 16*d^3* x^6) + a^3*(315*c^4*x + 840*c^3*d*x^3 + 1008*c^2*d^2*x^5 + 576*c*d^3*x^7 + 128*d^4*x^9))/(315*c^5*(c + d*x^2)^(9/2))
Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {296, 292, 292, 292, 208}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {(9 b c-8 a d) \int \frac {\left (b x^2+a\right )^3}{\left (d x^2+c\right )^{9/2}}dx}{9 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {(9 b c-8 a d) \left (\frac {6 a \int \frac {\left (b x^2+a\right )^2}{\left (d x^2+c\right )^{7/2}}dx}{7 c}+\frac {x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2}}\right )}{9 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {(9 b c-8 a d) \left (\frac {6 a \left (\frac {4 a \int \frac {b x^2+a}{\left (d x^2+c\right )^{5/2}}dx}{5 c}+\frac {x \left (a+b x^2\right )^2}{5 c \left (c+d x^2\right )^{5/2}}\right )}{7 c}+\frac {x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2}}\right )}{9 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {(9 b c-8 a d) \left (\frac {6 a \left (\frac {4 a \left (\frac {2 a \int \frac {1}{\left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \left (a+b x^2\right )}{3 c \left (c+d x^2\right )^{3/2}}\right )}{5 c}+\frac {x \left (a+b x^2\right )^2}{5 c \left (c+d x^2\right )^{5/2}}\right )}{7 c}+\frac {x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2}}\right )}{9 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {(9 b c-8 a d) \left (\frac {6 a \left (\frac {4 a \left (\frac {x \left (a+b x^2\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 a x}{3 c^2 \sqrt {c+d x^2}}\right )}{5 c}+\frac {x \left (a+b x^2\right )^2}{5 c \left (c+d x^2\right )^{5/2}}\right )}{7 c}+\frac {x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2}}\right )}{9 c (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)}\) |
Input:
Int[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]
Output:
-1/9*(d*x*(a + b*x^2)^4)/(c*(b*c - a*d)*(c + d*x^2)^(9/2)) + ((9*b*c - 8*a *d)*((x*(a + b*x^2)^3)/(7*c*(c + d*x^2)^(7/2)) + (6*a*((x*(a + b*x^2)^2)/( 5*c*(c + d*x^2)^(5/2)) + (4*a*((x*(a + b*x^2))/(3*c*(c + d*x^2)^(3/2)) + ( 2*a*x)/(3*c^2*Sqrt[c + d*x^2])))/(5*c)))/(7*c)))/(9*c*(b*c - a*d))
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Time = 0.46 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.61
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (\frac {1}{7} b^{3} x^{6}+\frac {3}{5} a \,b^{2} x^{4}+a^{2} b \,x^{2}+a^{3}\right ) c^{4}+\frac {8 d \,x^{2} \left (\frac {1}{84} b^{3} x^{6}+\frac {9}{70} a \,b^{2} x^{4}+\frac {9}{20} a^{2} b \,x^{2}+a^{3}\right ) c^{3}}{3}+\frac {16 a \left (\frac {1}{42} b^{2} x^{4}+\frac {3}{14} a b \,x^{2}+a^{2}\right ) d^{2} x^{4} c^{2}}{5}+\frac {64 a^{2} d^{3} \left (\frac {b \,x^{2}}{12}+a \right ) x^{6} c}{35}+\frac {128 a^{3} d^{4} x^{8}}{315}\right ) x}{\left (x^{2} d +c \right )^{\frac {9}{2}} c^{5}}\) | \(150\) |
| gosper | \(\frac {x \left (128 a^{3} d^{4} x^{8}+48 a^{2} b c \,d^{3} x^{8}+24 a \,b^{2} c^{2} d^{2} x^{8}+10 b^{3} c^{3} d \,x^{8}+576 a^{3} c \,d^{3} x^{6}+216 a^{2} b \,c^{2} d^{2} x^{6}+108 a \,b^{2} c^{3} d \,x^{6}+45 b^{3} c^{4} x^{6}+1008 a^{3} c^{2} d^{2} x^{4}+378 a^{2} b \,c^{3} d \,x^{4}+189 a \,b^{2} c^{4} x^{4}+840 a^{3} c^{3} d \,x^{2}+315 a^{2} b \,c^{4} x^{2}+315 a^{3} c^{4}\right )}{315 \left (x^{2} d +c \right )^{\frac {9}{2}} c^{5}}\) | \(190\) |
| trager | \(\frac {x \left (128 a^{3} d^{4} x^{8}+48 a^{2} b c \,d^{3} x^{8}+24 a \,b^{2} c^{2} d^{2} x^{8}+10 b^{3} c^{3} d \,x^{8}+576 a^{3} c \,d^{3} x^{6}+216 a^{2} b \,c^{2} d^{2} x^{6}+108 a \,b^{2} c^{3} d \,x^{6}+45 b^{3} c^{4} x^{6}+1008 a^{3} c^{2} d^{2} x^{4}+378 a^{2} b \,c^{3} d \,x^{4}+189 a \,b^{2} c^{4} x^{4}+840 a^{3} c^{3} d \,x^{2}+315 a^{2} b \,c^{4} x^{2}+315 a^{3} c^{4}\right )}{315 \left (x^{2} d +c \right )^{\frac {9}{2}} c^{5}}\) | \(190\) |
| orering | \(\frac {x \left (128 a^{3} d^{4} x^{8}+48 a^{2} b c \,d^{3} x^{8}+24 a \,b^{2} c^{2} d^{2} x^{8}+10 b^{3} c^{3} d \,x^{8}+576 a^{3} c \,d^{3} x^{6}+216 a^{2} b \,c^{2} d^{2} x^{6}+108 a \,b^{2} c^{3} d \,x^{6}+45 b^{3} c^{4} x^{6}+1008 a^{3} c^{2} d^{2} x^{4}+378 a^{2} b \,c^{3} d \,x^{4}+189 a \,b^{2} c^{4} x^{4}+840 a^{3} c^{3} d \,x^{2}+315 a^{2} b \,c^{4} x^{2}+315 a^{3} c^{4}\right )}{315 \left (x^{2} d +c \right )^{\frac {9}{2}} c^{5}}\) | \(190\) |
| default | \(a^{3} \left (\frac {x}{9 c \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 c \left (x^{2} d +c \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 c \left (x^{2} d +c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (x^{2} d +c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {x^{2} d +c}}\right )}{7 c}\right )}{9 c}}{c}\right )+b^{3} \left (-\frac {x^{5}}{4 d \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {5 c \left (-\frac {x^{3}}{6 d \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {c \left (-\frac {x}{8 d \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {c \left (\frac {x}{9 c \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 c \left (x^{2} d +c \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 c \left (x^{2} d +c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (x^{2} d +c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {x^{2} d +c}}\right )}{7 c}\right )}{9 c}}{c}\right )}{8 d}\right )}{2 d}\right )}{4 d}\right )+3 a \,b^{2} \left (-\frac {x^{3}}{6 d \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {c \left (-\frac {x}{8 d \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {c \left (\frac {x}{9 c \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 c \left (x^{2} d +c \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 c \left (x^{2} d +c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (x^{2} d +c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {x^{2} d +c}}\right )}{7 c}\right )}{9 c}}{c}\right )}{8 d}\right )}{2 d}\right )+3 a^{2} b \left (-\frac {x}{8 d \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {c \left (\frac {x}{9 c \left (x^{2} d +c \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 c \left (x^{2} d +c \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 c \left (x^{2} d +c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (x^{2} d +c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {x^{2} d +c}}\right )}{7 c}\right )}{9 c}}{c}\right )}{8 d}\right )\) | \(536\) |
Input:
int((b*x^2+a)^3/(d*x^2+c)^(11/2),x,method=_RETURNVERBOSE)
Output:
((1/7*b^3*x^6+3/5*a*b^2*x^4+a^2*b*x^2+a^3)*c^4+8/3*d*x^2*(1/84*b^3*x^6+9/7 0*a*b^2*x^4+9/20*a^2*b*x^2+a^3)*c^3+16/5*a*(1/42*b^2*x^4+3/14*a*b*x^2+a^2) *d^2*x^4*c^2+64/35*a^2*d^3*(1/12*b*x^2+a)*x^6*c+128/315*a^3*d^4*x^8)/(d*x^ 2+c)^(9/2)*x/c^5
Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=\frac {{\left (2 \, {\left (5 \, b^{3} c^{3} d + 12 \, a b^{2} c^{2} d^{2} + 24 \, a^{2} b c d^{3} + 64 \, a^{3} d^{4}\right )} x^{9} + 315 \, a^{3} c^{4} x + 9 \, {\left (5 \, b^{3} c^{4} + 12 \, a b^{2} c^{3} d + 24 \, a^{2} b c^{2} d^{2} + 64 \, a^{3} c d^{3}\right )} x^{7} + 63 \, {\left (3 \, a b^{2} c^{4} + 6 \, a^{2} b c^{3} d + 16 \, a^{3} c^{2} d^{2}\right )} x^{5} + 105 \, {\left (3 \, a^{2} b c^{4} + 8 \, a^{3} c^{3} d\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{315 \, {\left (c^{5} d^{5} x^{10} + 5 \, c^{6} d^{4} x^{8} + 10 \, c^{7} d^{3} x^{6} + 10 \, c^{8} d^{2} x^{4} + 5 \, c^{9} d x^{2} + c^{10}\right )}} \] Input:
integrate((b*x^2+a)^3/(d*x^2+c)^(11/2),x, algorithm="fricas")
Output:
1/315*(2*(5*b^3*c^3*d + 12*a*b^2*c^2*d^2 + 24*a^2*b*c*d^3 + 64*a^3*d^4)*x^ 9 + 315*a^3*c^4*x + 9*(5*b^3*c^4 + 12*a*b^2*c^3*d + 24*a^2*b*c^2*d^2 + 64* a^3*c*d^3)*x^7 + 63*(3*a*b^2*c^4 + 6*a^2*b*c^3*d + 16*a^3*c^2*d^2)*x^5 + 1 05*(3*a^2*b*c^4 + 8*a^3*c^3*d)*x^3)*sqrt(d*x^2 + c)/(c^5*d^5*x^10 + 5*c^6* d^4*x^8 + 10*c^7*d^3*x^6 + 10*c^8*d^2*x^4 + 5*c^9*d*x^2 + c^10)
\[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{3}}{\left (c + d x^{2}\right )^{\frac {11}{2}}}\, dx \] Input:
integrate((b*x**2+a)**3/(d*x**2+c)**(11/2),x)
Output:
Integral((a + b*x**2)**3/(c + d*x**2)**(11/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (225) = 450\).
Time = 0.04 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=-\frac {b^{3} x^{5}}{4 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d} - \frac {5 \, b^{3} c x^{3}}{24 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d^{2}} - \frac {a b^{2} x^{3}}{2 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d} + \frac {128 \, a^{3} x}{315 \, \sqrt {d x^{2} + c} c^{5}} + \frac {64 \, a^{3} x}{315 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{4}} + \frac {16 \, a^{3} x}{105 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{3}} + \frac {8 \, a^{3} x}{63 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} c^{2}} + \frac {a^{3} x}{9 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} c} + \frac {b^{3} x}{84 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} d^{3}} + \frac {2 \, b^{3} x}{63 \, \sqrt {d x^{2} + c} c^{2} d^{3}} + \frac {b^{3} x}{63 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c d^{3}} + \frac {5 \, b^{3} c x}{504 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d^{3}} - \frac {5 \, b^{3} c^{2} x}{72 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d^{3}} + \frac {a b^{2} x}{42 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d^{2}} + \frac {8 \, a b^{2} x}{105 \, \sqrt {d x^{2} + c} c^{3} d^{2}} + \frac {4 \, a b^{2} x}{105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2} d^{2}} + \frac {a b^{2} x}{35 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c d^{2}} - \frac {a b^{2} c x}{6 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d^{2}} - \frac {a^{2} b x}{3 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d} + \frac {16 \, a^{2} b x}{105 \, \sqrt {d x^{2} + c} c^{4} d} + \frac {8 \, a^{2} b x}{105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3} d} + \frac {2 \, a^{2} b x}{35 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{2} d} + \frac {a^{2} b x}{21 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} c d} \] Input:
integrate((b*x^2+a)^3/(d*x^2+c)^(11/2),x, algorithm="maxima")
Output:
-1/4*b^3*x^5/((d*x^2 + c)^(9/2)*d) - 5/24*b^3*c*x^3/((d*x^2 + c)^(9/2)*d^2 ) - 1/2*a*b^2*x^3/((d*x^2 + c)^(9/2)*d) + 128/315*a^3*x/(sqrt(d*x^2 + c)*c ^5) + 64/315*a^3*x/((d*x^2 + c)^(3/2)*c^4) + 16/105*a^3*x/((d*x^2 + c)^(5/ 2)*c^3) + 8/63*a^3*x/((d*x^2 + c)^(7/2)*c^2) + 1/9*a^3*x/((d*x^2 + c)^(9/2 )*c) + 1/84*b^3*x/((d*x^2 + c)^(5/2)*d^3) + 2/63*b^3*x/(sqrt(d*x^2 + c)*c^ 2*d^3) + 1/63*b^3*x/((d*x^2 + c)^(3/2)*c*d^3) + 5/504*b^3*c*x/((d*x^2 + c) ^(7/2)*d^3) - 5/72*b^3*c^2*x/((d*x^2 + c)^(9/2)*d^3) + 1/42*a*b^2*x/((d*x^ 2 + c)^(7/2)*d^2) + 8/105*a*b^2*x/(sqrt(d*x^2 + c)*c^3*d^2) + 4/105*a*b^2* x/((d*x^2 + c)^(3/2)*c^2*d^2) + 1/35*a*b^2*x/((d*x^2 + c)^(5/2)*c*d^2) - 1 /6*a*b^2*c*x/((d*x^2 + c)^(9/2)*d^2) - 1/3*a^2*b*x/((d*x^2 + c)^(9/2)*d) + 16/105*a^2*b*x/(sqrt(d*x^2 + c)*c^4*d) + 8/105*a^2*b*x/((d*x^2 + c)^(3/2) *c^3*d) + 2/35*a^2*b*x/((d*x^2 + c)^(5/2)*c^2*d) + 1/21*a^2*b*x/((d*x^2 + c)^(7/2)*c*d)
Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=\frac {{\left ({\left ({\left (x^{2} {\left (\frac {2 \, {\left (5 \, b^{3} c^{3} d^{5} + 12 \, a b^{2} c^{2} d^{6} + 24 \, a^{2} b c d^{7} + 64 \, a^{3} d^{8}\right )} x^{2}}{c^{5} d^{4}} + \frac {9 \, {\left (5 \, b^{3} c^{4} d^{4} + 12 \, a b^{2} c^{3} d^{5} + 24 \, a^{2} b c^{2} d^{6} + 64 \, a^{3} c d^{7}\right )}}{c^{5} d^{4}}\right )} + \frac {63 \, {\left (3 \, a b^{2} c^{4} d^{4} + 6 \, a^{2} b c^{3} d^{5} + 16 \, a^{3} c^{2} d^{6}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac {105 \, {\left (3 \, a^{2} b c^{4} d^{4} + 8 \, a^{3} c^{3} d^{5}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac {315 \, a^{3}}{c}\right )} x}{315 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}}} \] Input:
integrate((b*x^2+a)^3/(d*x^2+c)^(11/2),x, algorithm="giac")
Output:
1/315*(((x^2*(2*(5*b^3*c^3*d^5 + 12*a*b^2*c^2*d^6 + 24*a^2*b*c*d^7 + 64*a^ 3*d^8)*x^2/(c^5*d^4) + 9*(5*b^3*c^4*d^4 + 12*a*b^2*c^3*d^5 + 24*a^2*b*c^2* d^6 + 64*a^3*c*d^7)/(c^5*d^4)) + 63*(3*a*b^2*c^4*d^4 + 6*a^2*b*c^3*d^5 + 1 6*a^3*c^2*d^6)/(c^5*d^4))*x^2 + 105*(3*a^2*b*c^4*d^4 + 8*a^3*c^3*d^5)/(c^5 *d^4))*x^2 + 315*a^3/c)*x/(d*x^2 + c)^(9/2)
Time = 0.75 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=\frac {x\,\left (\frac {a^3}{9\,c}-\frac {c\,\left (\frac {c\,\left (\frac {b^3}{9\,d}-\frac {a\,b^2}{3\,c}\right )}{d}+\frac {a^2\,b}{3\,c}\right )}{d}\right )}{{\left (d\,x^2+c\right )}^{9/2}}-\frac {x\,\left (\frac {b^3}{5\,d^3}-\frac {16\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-4\,b^3\,c^3}{105\,c^3\,d^3}\right )}{{\left (d\,x^2+c\right )}^{5/2}}+\frac {x\,\left (\frac {c\,\left (\frac {b^3}{7\,d^2}-\frac {b^2\,\left (3\,a\,d-b\,c\right )}{7\,c\,d^2}\right )}{d}+\frac {8\,a^3\,d^3+3\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d+b^3\,c^3}{63\,c^2\,d^3}\right )}{{\left (d\,x^2+c\right )}^{7/2}}+\frac {x\,\left (64\,a^3\,d^3+24\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{315\,c^4\,d^3\,{\left (d\,x^2+c\right )}^{3/2}}+\frac {x\,\left (128\,a^3\,d^3+48\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+10\,b^3\,c^3\right )}{315\,c^5\,d^3\,\sqrt {d\,x^2+c}} \] Input:
int((a + b*x^2)^3/(c + d*x^2)^(11/2),x)
Output:
(x*(a^3/(9*c) - (c*((c*(b^3/(9*d) - (a*b^2)/(3*c)))/d + (a^2*b)/(3*c)))/d) )/(c + d*x^2)^(9/2) - (x*(b^3/(5*d^3) - (16*a^3*d^3 - 4*b^3*c^3 + 3*a*b^2* c^2*d + 6*a^2*b*c*d^2)/(105*c^3*d^3)))/(c + d*x^2)^(5/2) + (x*((c*(b^3/(7* d^2) - (b^2*(3*a*d - b*c))/(7*c*d^2)))/d + (8*a^3*d^3 + b^3*c^3 - 3*a*b^2* c^2*d + 3*a^2*b*c*d^2)/(63*c^2*d^3)))/(c + d*x^2)^(7/2) + (x*(64*a^3*d^3 + 5*b^3*c^3 + 12*a*b^2*c^2*d + 24*a^2*b*c*d^2))/(315*c^4*d^3*(c + d*x^2)^(3 /2)) + (x*(128*a^3*d^3 + 10*b^3*c^3 + 24*a*b^2*c^2*d + 48*a^2*b*c*d^2))/(3 15*c^5*d^3*(c + d*x^2)^(1/2))
Time = 0.23 (sec) , antiderivative size = 738, normalized size of antiderivative = 3.01 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx=\frac {128 \sqrt {d \,x^{2}+c}\, a^{3} d^{8} x^{9}-128 \sqrt {d}\, a^{3} c^{5} d^{3}-128 \sqrt {d}\, a^{3} d^{8} x^{10}-10 \sqrt {d}\, b^{3} c^{3} d^{5} x^{10}-480 \sqrt {d}\, a^{2} b \,c^{3} d^{5} x^{6}-240 \sqrt {d}\, a^{2} b \,c^{2} d^{6} x^{8}-48 \sqrt {d}\, a^{2} b c \,d^{7} x^{10}-120 \sqrt {d}\, a \,b^{2} c^{6} d^{2} x^{2}-240 \sqrt {d}\, a \,b^{2} c^{5} d^{3} x^{4}-240 \sqrt {d}\, a \,b^{2} c^{4} d^{4} x^{6}-120 \sqrt {d}\, a \,b^{2} c^{3} d^{5} x^{8}-24 \sqrt {d}\, a \,b^{2} c^{2} d^{6} x^{10}-10 \sqrt {d}\, b^{3} c^{8}+315 \sqrt {d \,x^{2}+c}\, a^{2} b \,c^{4} d^{4} x^{3}+378 \sqrt {d \,x^{2}+c}\, a^{2} b \,c^{3} d^{5} x^{5}+216 \sqrt {d \,x^{2}+c}\, a^{2} b \,c^{2} d^{6} x^{7}+48 \sqrt {d \,x^{2}+c}\, a^{2} b c \,d^{7} x^{9}+189 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{4} d^{4} x^{5}+108 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{3} d^{5} x^{7}+24 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{2} d^{6} x^{9}-240 \sqrt {d}\, a^{2} b \,c^{5} d^{3} x^{2}-480 \sqrt {d}\, a^{2} b \,c^{4} d^{4} x^{4}+315 \sqrt {d \,x^{2}+c}\, a^{3} c^{4} d^{4} x +840 \sqrt {d \,x^{2}+c}\, a^{3} c^{3} d^{5} x^{3}+1008 \sqrt {d \,x^{2}+c}\, a^{3} c^{2} d^{6} x^{5}+576 \sqrt {d \,x^{2}+c}\, a^{3} c \,d^{7} x^{7}+45 \sqrt {d \,x^{2}+c}\, b^{3} c^{4} d^{4} x^{7}+10 \sqrt {d \,x^{2}+c}\, b^{3} c^{3} d^{5} x^{9}-640 \sqrt {d}\, a^{3} c^{4} d^{4} x^{2}-1280 \sqrt {d}\, a^{3} c^{3} d^{5} x^{4}-1280 \sqrt {d}\, a^{3} c^{2} d^{6} x^{6}-640 \sqrt {d}\, a^{3} c \,d^{7} x^{8}-48 \sqrt {d}\, a^{2} b \,c^{6} d^{2}-24 \sqrt {d}\, a \,b^{2} c^{7} d -50 \sqrt {d}\, b^{3} c^{7} d \,x^{2}-100 \sqrt {d}\, b^{3} c^{6} d^{2} x^{4}-100 \sqrt {d}\, b^{3} c^{5} d^{3} x^{6}-50 \sqrt {d}\, b^{3} c^{4} d^{4} x^{8}}{315 c^{5} d^{4} \left (d^{5} x^{10}+5 c \,d^{4} x^{8}+10 c^{2} d^{3} x^{6}+10 c^{3} d^{2} x^{4}+5 c^{4} d \,x^{2}+c^{5}\right )} \] Input:
int((b*x^2+a)^3/(d*x^2+c)^(11/2),x)
Output:
(315*sqrt(c + d*x**2)*a**3*c**4*d**4*x + 840*sqrt(c + d*x**2)*a**3*c**3*d* *5*x**3 + 1008*sqrt(c + d*x**2)*a**3*c**2*d**6*x**5 + 576*sqrt(c + d*x**2) *a**3*c*d**7*x**7 + 128*sqrt(c + d*x**2)*a**3*d**8*x**9 + 315*sqrt(c + d*x **2)*a**2*b*c**4*d**4*x**3 + 378*sqrt(c + d*x**2)*a**2*b*c**3*d**5*x**5 + 216*sqrt(c + d*x**2)*a**2*b*c**2*d**6*x**7 + 48*sqrt(c + d*x**2)*a**2*b*c* d**7*x**9 + 189*sqrt(c + d*x**2)*a*b**2*c**4*d**4*x**5 + 108*sqrt(c + d*x* *2)*a*b**2*c**3*d**5*x**7 + 24*sqrt(c + d*x**2)*a*b**2*c**2*d**6*x**9 + 45 *sqrt(c + d*x**2)*b**3*c**4*d**4*x**7 + 10*sqrt(c + d*x**2)*b**3*c**3*d**5 *x**9 - 128*sqrt(d)*a**3*c**5*d**3 - 640*sqrt(d)*a**3*c**4*d**4*x**2 - 128 0*sqrt(d)*a**3*c**3*d**5*x**4 - 1280*sqrt(d)*a**3*c**2*d**6*x**6 - 640*sqr t(d)*a**3*c*d**7*x**8 - 128*sqrt(d)*a**3*d**8*x**10 - 48*sqrt(d)*a**2*b*c* *6*d**2 - 240*sqrt(d)*a**2*b*c**5*d**3*x**2 - 480*sqrt(d)*a**2*b*c**4*d**4 *x**4 - 480*sqrt(d)*a**2*b*c**3*d**5*x**6 - 240*sqrt(d)*a**2*b*c**2*d**6*x **8 - 48*sqrt(d)*a**2*b*c*d**7*x**10 - 24*sqrt(d)*a*b**2*c**7*d - 120*sqrt (d)*a*b**2*c**6*d**2*x**2 - 240*sqrt(d)*a*b**2*c**5*d**3*x**4 - 240*sqrt(d )*a*b**2*c**4*d**4*x**6 - 120*sqrt(d)*a*b**2*c**3*d**5*x**8 - 24*sqrt(d)*a *b**2*c**2*d**6*x**10 - 10*sqrt(d)*b**3*c**8 - 50*sqrt(d)*b**3*c**7*d*x**2 - 100*sqrt(d)*b**3*c**6*d**2*x**4 - 100*sqrt(d)*b**3*c**5*d**3*x**6 - 50* sqrt(d)*b**3*c**4*d**4*x**8 - 10*sqrt(d)*b**3*c**3*d**5*x**10)/(315*c**5*d **4*(c**5 + 5*c**4*d*x**2 + 10*c**3*d**2*x**4 + 10*c**2*d**3*x**6 + 5*c...