Integrand size = 25, antiderivative size = 527 \[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=-\frac {4 \left (8 b^2 c^4-9 a b c^2 d^2+25 a^2 d^4\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{1155 b^2 d^3}-\frac {4 c \left (8 b c^2+a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{1155 b d^3}+\frac {2 (c+d x)^{5/2} \left (8 b c^2+15 a d^2-14 b c d x\right ) \sqrt {a-b x^2}}{231 b d^3}-\frac {2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}{11 b d}-\frac {4 \sqrt {a} c \left (8 b^2 c^4-15 a b c^2 d^2+103 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{1155 b^{3/2} d^4 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} \left (8 b^3 c^6-17 a b^2 c^4 d^2+34 a^2 b c^2 d^4-25 a^3 d^6\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{1155 b^{5/2} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-4/1155*(25*a^2*d^4-9*a*b*c^2*d^2+8*b^2*c^4)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2 )/b^2/d^3-4/1155*c*(a*d^2+8*b*c^2)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b/d^3+2/ 231*(d*x+c)^(5/2)*(-14*b*c*d*x+15*a*d^2+8*b*c^2)*(-b*x^2+a)^(1/2)/b/d^3-2/ 11*(d*x+c)^(5/2)*(-b*x^2+a)^(3/2)/b/d-4/1155*a^(1/2)*c*(103*a^2*d^4-15*a*b *c^2*d^2+8*b^2*c^4)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/ 2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2 ))/b^(3/2)/d^4/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1 /2)+4/1155*a^(1/2)*(-25*a^3*d^6+34*a^2*b*c^2*d^4-17*a*b^2*c^4*d^2+8*b^3*c^ 6)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*Ellipti cF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a ^(1/2)*d))^(1/2))/b^(5/2)/d^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.01 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.25 \[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\frac {2 \sqrt {a-b x^2} \left ((c+d x) \left (-50 a^2 d^4-2 a b d^2 \left (7 c^2+31 c d x+15 d^2 x^2\right )+b^2 \left (8 c^4-6 c^3 d x+5 c^2 d^2 x^2+140 c d^3 x^3+105 d^4 x^4\right )\right )-\frac {2 \left (c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (8 b^2 c^4-15 a b c^2 d^2+103 a^2 d^4\right ) \left (a-b x^2\right )+i \sqrt {b} c \left (8 b^{5/2} c^5-8 \sqrt {a} b^2 c^4 d-15 a b^{3/2} c^3 d^2+15 a^{3/2} b c^2 d^3+103 a^2 \sqrt {b} c d^4-103 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (-8 b^{5/2} c^5+2 \sqrt {a} b^2 c^4 d+15 a b^{3/2} c^3 d^2+69 a^{3/2} b c^2 d^3-103 a^2 \sqrt {b} c d^4+25 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{1155 b^2 d^3 \sqrt {c+d x}} \] Input:
Integrate[x^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*((c + d*x)*(-50*a^2*d^4 - 2*a*b*d^2*(7*c^2 + 31*c*d*x + 15*d^2*x^2) + b^2*(8*c^4 - 6*c^3*d*x + 5*c^2*d^2*x^2 + 140*c*d^3*x^3 + 10 5*d^4*x^4)) - (2*(c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(8*b^2*c^4 - 15*a*b *c^2*d^2 + 103*a^2*d^4)*(a - b*x^2) + I*Sqrt[b]*c*(8*b^(5/2)*c^5 - 8*Sqrt[ a]*b^2*c^4*d - 15*a*b^(3/2)*c^3*d^2 + 15*a^(3/2)*b*c^2*d^3 + 103*a^2*Sqrt[ b]*c*d^4 - 103*a^(5/2)*d^5)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt [-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*Arc Sinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d )/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*d*(-8*b^(5/2)*c^5 + 2*Sqrt[a]*b^2*c ^4*d + 15*a*b^(3/2)*c^3*d^2 + 69*a^(3/2)*b*c^2*d^3 - 103*a^2*Sqrt[b]*c*d^4 + 25*a^(5/2)*d^5)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt [a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt [-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b ]*c - Sqrt[a]*d)]))/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(11 55*b^2*d^3*Sqrt[c + d*x])
Time = 0.81 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {604, 27, 687, 27, 687, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int x^2 \sqrt {a-b x^2} (c+d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle -\frac {2 \int -\frac {1}{2} d (5 a d-6 b c x) (c+d x)^{3/2} \sqrt {a-b x^2}dx}{11 b d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (5 a d-6 b c x) (c+d x)^{3/2} \sqrt {a-b x^2}dx}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}-\frac {2 \int -\frac {9}{2} b \sqrt {c+d x} \left (3 a c d-\left (2 b c^2-5 a d^2\right ) x\right ) \sqrt {a-b x^2}dx}{9 b}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {c+d x} \left (3 a c d-\left (2 b c^2-5 a d^2\right ) x\right ) \sqrt {a-b x^2}dx+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {\left (a d \left (19 b c^2+5 a d^2\right )-2 b c \left (b c^2-13 a d^2\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a d \left (19 b c^2+5 a d^2\right )-2 b c \left (b c^2-13 a d^2\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (2 b^2 c^4+69 a b d^2 c^2+25 a^2 d^4\right )+b c \left (8 b^2 c^4-15 a b d^2 c^2+103 a^2 d^4\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {a d \left (2 b^2 c^4+69 a b d^2 c^2+25 a^2 d^4\right )+b c \left (8 b^2 c^4-15 a b d^2 c^2+103 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {b c \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (25 a^2 d^4-9 a b c^2 d^2+8 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (103 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a^2 d^4-6 b c d x \left (b c^2-13 a d^2\right )-9 a b c^2 d^2+8 b^2 c^4\right )}{15 d^2}}{7 b}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (2 b c^2-5 a d^2\right )}{7 b}+\frac {4}{3} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}{11 b d}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b d}\) |
Input:
Int[x^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2],x]
Output:
(-2*(c + d*x)^(5/2)*(a - b*x^2)^(3/2))/(11*b*d) + ((2*(2*b*c^2 - 5*a*d^2)* Sqrt[c + d*x]*(a - b*x^2)^(3/2))/(7*b) + (4*c*(c + d*x)^(3/2)*(a - b*x^2)^ (3/2))/3 + ((2*Sqrt[c + d*x]*(8*b^2*c^4 - 9*a*b*c^2*d^2 + 25*a^2*d^4 - 6*b *c*d*(b*c^2 - 13*a*d^2)*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((-2*Sqrt[a]*Sqr t[b]*c*(8*b^2*c^4 - 15*a*b*c^2*d^2 + 103*a^2*d^4)*Sqrt[c + d*x]*Sqrt[1 - ( b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/( (Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a ]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(8*b^2*c^4 - 9*a*b*c^2 *d^2 + 25*a^2*d^4)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[ 1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2 *d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])) )/(15*d^2))/(7*b))/(11*b*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(b*d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b* x^2)^p*ExpandToSum[b*d^m*(m + n + 2*p + 1)*x^m - b*(m + n + 2*p + 1)*(c + d *x)^m - (c + d*x)^(m - 2)*(a*d^2*(m + n - 1) - b*c^2*(m + n + 2*p + 1) - 2* b*c*d*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(445)=890\).
Time = 2.26 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.77
| method | result | size |
| risch | \(-\frac {2 \left (-105 b^{2} d^{4} x^{4}-140 b^{2} c \,d^{3} x^{3}+30 a b \,d^{4} x^{2}-5 d^{2} c^{2} x^{2} b^{2}+62 a b c \,d^{3} x +6 b^{2} c^{3} d x +50 a^{2} d^{4}+14 b \,c^{2} d^{2} a -8 b^{2} c^{4}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{1155 b^{2} d^{3}}+\frac {2 \left (\frac {c \left (103 a^{2} d^{4}-15 b \,c^{2} d^{2} a +8 b^{2} c^{4}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {25 a^{3} d^{5} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 a b \,c^{4} d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {69 a^{2} c^{2} d^{3} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{1155 b^{2} d^{3} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(933\) |
| elliptic | \(\text {Expression too large to display}\) | \(1009\) |
| default | \(\text {Expression too large to display}\) | \(1881\) |
Input:
int(x^2*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/1155*(-105*b^2*d^4*x^4-140*b^2*c*d^3*x^3+30*a*b*d^4*x^2-5*b^2*c^2*d^2*x ^2+62*a*b*c*d^3*x+6*b^2*c^3*d*x+50*a^2*d^4+14*a*b*c^2*d^2-8*b^2*c^4)/b^2*( d*x+c)^(1/2)/d^3*(-b*x^2+a)^(1/2)+2/1155/b^2/d^3*(c*(103*a^2*d^4-15*a*b*c^ 2*d^2+8*b^2*c^4)*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^( 1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b) ^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((c/d-1/b*(a*b)^(1/2))*El lipticE(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^ (1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b )^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/ 2)))+25*a^3*d^5/b*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^ (1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b )^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*(( x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^( 1/2)))^(1/2))+2*a*b*c^4*d*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b) ^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2) )*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^ (1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b *(a*b)^(1/2)))^(1/2))+69*a^2*c^2*d^3*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/ 2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b* (a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*El...
Time = 0.12 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.70 \[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\frac {2 \, {\left (2 \, {\left (8 \, b^{3} c^{6} - 21 \, a b^{2} c^{4} d^{2} - 104 \, a^{2} b c^{2} d^{4} - 75 \, a^{3} d^{6}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (8 \, b^{3} c^{5} d - 15 \, a b^{2} c^{3} d^{3} + 103 \, a^{2} b c d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (105 \, b^{3} d^{6} x^{4} + 140 \, b^{3} c d^{5} x^{3} + 8 \, b^{3} c^{4} d^{2} - 14 \, a b^{2} c^{2} d^{4} - 50 \, a^{2} b d^{6} + 5 \, {\left (b^{3} c^{2} d^{4} - 6 \, a b^{2} d^{6}\right )} x^{2} - 2 \, {\left (3 \, b^{3} c^{3} d^{3} + 31 \, a b^{2} c d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{3465 \, b^{3} d^{5}} \] Input:
integrate(x^2*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/3465*(2*(8*b^3*c^6 - 21*a*b^2*c^4*d^2 - 104*a^2*b*c^2*d^4 - 75*a^3*d^6)* sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 6*(8*b^3*c^5*d - 15*a*b^2*c^3* d^3 + 103*a^2*b*c*d^5)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b *d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(105*b^3*d^6*x^4 + 140*b^3*c*d^5*x^3 + 8*b^3*c^4*d^2 - 14*a*b^2*c^2*d^ 4 - 50*a^2*b*d^6 + 5*(b^3*c^2*d^4 - 6*a*b^2*d^6)*x^2 - 2*(3*b^3*c^3*d^3 + 31*a*b^2*c*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^5)
\[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\int x^{2} \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(d*x+c)**(3/2)*(-b*x**2+a)**(1/2),x)
Output:
Integral(x**2*sqrt(a - b*x**2)*(c + d*x)**(3/2), x)
\[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)*x^2, x)
\[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)*x^2, x)
Timed out. \[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\int x^2\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2} \,d x \] Input:
int(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(3/2),x)
Output:
int(x^2*(a - b*x^2)^(1/2)*(c + d*x)^(3/2), x)
\[ \int x^2 (c+d x)^{3/2} \sqrt {a-b x^2} \, dx=\frac {-306 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} d^{3}+2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,c^{2} d -124 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b c \,d^{2} x -60 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,d^{3} x^{2}-12 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{3} x +10 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{2} d \,x^{2}+280 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c \,d^{2} x^{3}+210 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} d^{3} x^{4}-309 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{4}+45 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{2} d^{2}-24 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c^{4}+153 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} d^{4}+123 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,c^{2} d^{2}+12 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{4}}{1155 b^{2} d^{2}} \] Input:
int(x^2*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2),x)
Output:
( - 306*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 + 2*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2*d - 124*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x - 6 0*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*d**3*x**2 - 12*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*x + 10*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x**2 + 280*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**2*x**3 + 210*sqrt(c + d*x)* sqrt(a - b*x**2)*b**2*d**3*x**4 - 309*int((sqrt(c + d*x)*sqrt(a - b*x**2)* x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*d**4 + 45*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b** 2*c**2*d**2 - 24*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**3*c**4 + 153*int((sqrt(c + d*x)*sqrt(a - b*x**2 ))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**3*d**4 + 123*int((sqrt(c + d* x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*c**2*d* *2 + 12*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d *x**3),x)*a*b**2*c**4)/(1155*b**2*d**2)