Integrand size = 23, antiderivative size = 516 \[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\frac {4 \left (4 b^2 c^4-21 a b c^2 d^2-15 a^2 d^4\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{693 b^2 d^2}-\frac {16}{693} c \left (\frac {4 a}{b}-\frac {c^2}{d^2}\right ) (c+d x)^{3/2} \sqrt {a-b x^2}-\frac {4}{693} \left (\frac {9 a}{b}-\frac {4 c^2}{d^2}\right ) (c+d x)^{5/2} \sqrt {a-b x^2}-\frac {2 (4 c-9 d x) (c+d x)^{7/2} \sqrt {a-b x^2}}{99 d^2}+\frac {16 \sqrt {a} c \left (b c^2-9 a d^2\right ) \left (b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{693 b^{3/2} d^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} \left (4 b^3 c^6-25 a b^2 c^4 d^2+6 a^2 b c^2 d^4+15 a^3 d^6\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{693 b^{5/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
4/693*(-15*a^2*d^4-21*a*b*c^2*d^2+4*b^2*c^4)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2 )/b^2/d^2-16/693*c*(4*a/b-c^2/d^2)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)-4/693*(9 *a/b-4*c^2/d^2)*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)-2/99*(-9*d*x+4*c)*(d*x+c)^( 7/2)*(-b*x^2+a)^(1/2)/d^2+16/693*a^(1/2)*c*(-9*a*d^2+b*c^2)*(3*a*d^2+b*c^2 )*(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^3/(b^ (1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-4/693*a^(1/2)* (15*a^3*d^6+6*a^2*b*c^2*d^4-25*a*b^2*c^4*d^2+4*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)*EllipticF(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^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.90 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.28 \[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\frac {2 \sqrt {a-b x^2} \left (-\left ((c+d x) \left (30 a^2 d^4+2 a b d^2 \left (46 c^2+34 c d x+9 d^2 x^2\right )+b^2 \left (4 c^4-3 c^3 d x-113 c^2 d^2 x^2-161 c d^3 x^3-63 d^4 x^4\right )\right )\right )+\frac {2 \left (4 c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b^2 c^4-6 a b c^2 d^2-27 a^2 d^4\right ) \left (a-b x^2\right )+4 i \sqrt {b} c \left (b^{5/2} c^5-\sqrt {a} b^2 c^4 d-6 a b^{3/2} c^3 d^2+6 a^{3/2} b c^2 d^3-27 a^2 \sqrt {b} c d^4+27 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (4 b^{5/2} c^5-\sqrt {a} b^2 c^4 d-24 a b^{3/2} c^3 d^2+114 a^{3/2} b c^2 d^3-108 a^2 \sqrt {b} c d^4+15 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{693 b^2 d^2 \sqrt {c+d x}} \] Input:
Integrate[x*(c + d*x)^(5/2)*Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(-((c + d*x)*(30*a^2*d^4 + 2*a*b*d^2*(46*c^2 + 34*c*d*x + 9*d^2*x^2) + b^2*(4*c^4 - 3*c^3*d*x - 113*c^2*d^2*x^2 - 161*c*d^3*x^3 - 63*d^4*x^4))) + (2*(4*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b^2*c^4 - 6*a *b*c^2*d^2 - 27*a^2*d^4)*(a - b*x^2) + (4*I)*Sqrt[b]*c*(b^(5/2)*c^5 - Sqrt [a]*b^2*c^4*d - 6*a*b^(3/2)*c^3*d^2 + 6*a^(3/2)*b*c^2*d^3 - 27*a^2*Sqrt[b] *c*d^4 + 27*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*ArcSin h[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*(4*b^(5/2)*c^5 - Sqrt[a]*b^2*c^4*d - 24*a*b^(3/2)*c^3*d^2 + 114*a^(3/2)*b*c^2*d^3 - 108*a^2*Sqrt[b]*c*d^4 + 15 *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))))/(693*b^2 *d^2*Sqrt[c + d*x])
Time = 0.82 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {596, 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.
| \(\displaystyle \int x \sqrt {a-b x^2} (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 596 |
| \(\displaystyle \frac {5 \int (a d+b c x) (c+d x)^{3/2} \sqrt {a-b x^2}dx}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {5 \left (-\frac {2 \int -\frac {3}{2} b \sqrt {c+d x} \left (4 a c d+\left (b c^2+3 a d^2\right ) x\right ) \sqrt {a-b x^2}dx}{9 b}-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \int \sqrt {c+d x} \left (4 a c d+\left (b c^2+3 a d^2\right ) x\right ) \sqrt {a-b x^2}dx-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (-\frac {2 \int -\frac {\left (a d \left (29 b c^2+3 a d^2\right )+b c \left (b c^2+31 a d^2\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {\int \frac {\left (a d \left (29 b c^2+3 a d^2\right )+b c \left (b c^2+31 a d^2\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {4 \int \frac {b \left (a d \left (b^2 c^4-114 a b d^2 c^2-15 a^2 d^4\right )+4 b c \left (b c^2-9 a d^2\right ) \left (b c^2+3 a d^2\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \int \frac {a d \left (b^2 c^4-114 a b d^2 c^2-15 a^2 d^4\right )+4 b c \left (b c^2-9 a d^2\right ) \left (b c^2+3 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (\frac {4 b c \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (\frac {4 b c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {5 \left (\frac {1}{3} \left (\frac {-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-15 a^2 d^4-21 a b c^2 d^2+4 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-9 a d^2\right ) \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (-15 a^2 d^4-3 b c d x \left (31 a d^2+b c^2\right )-21 a b c^2 d^2+4 b^2 c^4\right )}{15 d^2}}{7 b}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (3 a d^2+b c^2\right )}{7 b}\right )-\frac {2}{9} c \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}\right )}{11 b}-\frac {2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}{11 b}\) |
Input:
Int[x*(c + d*x)^(5/2)*Sqrt[a - b*x^2],x]
Output:
(-2*(c + d*x)^(5/2)*(a - b*x^2)^(3/2))/(11*b) + (5*((-2*c*(c + d*x)^(3/2)* (a - b*x^2)^(3/2))/9 + ((-2*(b*c^2 + 3*a*d^2)*Sqrt[c + d*x]*(a - b*x^2)^(3 /2))/(7*b) + ((-2*Sqrt[c + d*x]*(4*b^2*c^4 - 21*a*b*c^2*d^2 - 15*a^2*d^4 - 3*b*c*d*(b*c^2 + 31*a*d^2)*x)*Sqrt[a - b*x^2])/(15*d^2) - (2*((-8*Sqrt[a] *Sqrt[b]*c*(b*c^2 - 9*a*d^2)*(b*c^2 + 3*a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x ^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sq rt[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)*(4*b^2*c^4 - 21*a*b*c^2*d ^2 - 15*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))/3))/(11*b)
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[(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[((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[((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. \(933\) vs. \(2(434)=868\).
Time = 2.31 (sec) , antiderivative size = 934, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(-\frac {2 \left (-63 b^{2} d^{4} x^{4}-161 b^{2} c \,d^{3} x^{3}+18 a b \,d^{4} x^{2}-113 d^{2} c^{2} x^{2} b^{2}+68 a b c \,d^{3} x -3 b^{2} c^{3} d x +30 a^{2} d^{4}+92 b \,c^{2} d^{2} a +4 b^{2} c^{4}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{693 d^{2} b^{2}}+\frac {2 \left (\frac {4 c \left (27 a^{2} d^{4}+6 b \,c^{2} d^{2} a -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 {15 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 {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 {114 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 )}}{693 b^{2} d^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(934\) |
| elliptic | \(\text {Expression too large to display}\) | \(1079\) |
| default | \(\text {Expression too large to display}\) | \(1880\) |
Input:
int(x*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/693/d^2*(-63*b^2*d^4*x^4-161*b^2*c*d^3*x^3+18*a*b*d^4*x^2-113*b^2*c^2*d ^2*x^2+68*a*b*c*d^3*x-3*b^2*c^3*d*x+30*a^2*d^4+92*a*b*c^2*d^2+4*b^2*c^4)/b ^2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)+2/693/b^2/d^2*(4*c*(27*a^2*d^4+6*a*b*c^2 *d^2-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))*Ellip ticE(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)) )+15*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))-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))+114*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)*Ellipt...
Time = 0.10 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.72 \[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, b^{3} c^{6} - 27 \, a b^{2} c^{4} d^{2} + 234 \, a^{2} b c^{2} d^{4} + 45 \, a^{3} d^{6}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 24 \, {\left (b^{3} c^{5} d - 6 \, a b^{2} c^{3} d^{3} - 27 \, a^{2} b c d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (63 \, b^{3} d^{6} x^{4} + 161 \, b^{3} c d^{5} x^{3} - 4 \, b^{3} c^{4} d^{2} - 92 \, a b^{2} c^{2} d^{4} - 30 \, a^{2} b d^{6} + {\left (113 \, b^{3} c^{2} d^{4} - 18 \, a b^{2} d^{6}\right )} x^{2} + {\left (3 \, b^{3} c^{3} d^{3} - 68 \, a b^{2} c d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{2079 \, b^{3} d^{4}} \] Input:
integrate(x*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/2079*(2*(4*b^3*c^6 - 27*a*b^2*c^4*d^2 + 234*a^2*b*c^2*d^4 + 45*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) + 24*(b^3*c^5*d - 6*a*b^2*c^3*d ^3 - 27*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*(63*b^3*d^6*x^4 + 161*b^3*c*d^5*x^3 - 4*b^3*c^4*d^2 - 92*a*b^2*c^2*d^4 - 30*a^2*b*d^6 + (113*b^3*c^2*d^4 - 18*a*b^2*d^6)*x^2 + (3*b^3*c^3*d^3 - 68 *a*b^2*c*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^4)
\[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int x \sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \] Input:
integrate(x*(d*x+c)**(5/2)*(-b*x**2+a)**(1/2),x)
Output:
Integral(x*sqrt(a - b*x**2)*(c + d*x)**(5/2), x)
\[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}} x \,d x } \] Input:
integrate(x*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)*x, x)
\[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}} x \,d x } \] Input:
integrate(x*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)*x, x)
Timed out. \[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\int x\,\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{5/2} \,d x \] Input:
int(x*(a - b*x^2)^(1/2)*(c + d*x)^(5/2),x)
Output:
int(x*(a - b*x^2)^(1/2)*(c + d*x)^(5/2), x)
\[ \int x (c+d x)^{5/2} \sqrt {a-b x^2} \, dx=\frac {-\frac {92 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} d^{3}}{231}-\frac {232 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,c^{2} d}{693}-\frac {136 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b c \,d^{2} x}{693}-\frac {4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,d^{3} x^{2}}{77}+\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{3} x}{231}+\frac {226 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{2} d \,x^{2}}{693}+\frac {46 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c \,d^{2} x^{3}}{99}+\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} d^{3} x^{4}}{11}-\frac {36 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{4}}{77}-\frac {8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{2} d^{2}}{77}+\frac {4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c^{4}}{231}+\frac {46 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} d^{4}}{231}+\frac {4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,c^{2} d^{2}}{11}-\frac {2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{4}}{231}}{b^{2} d} \] Input:
int(x*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2),x)
Output:
(2*( - 138*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 - 116*sqrt(c + d*x)*sq rt(a - b*x**2)*a*b*c**2*d - 68*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x - 18*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*d**3*x**2 + 3*sqrt(c + d*x)*sqrt( a - b*x**2)*b**2*c**3*x + 113*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x **2 + 161*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**2*x**3 + 63*sqrt(c + d* x)*sqrt(a - b*x**2)*b**2*d**3*x**4 - 162*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 - 36*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 + 6*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 + 69*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 + 126*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 - 3*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))/(693*b**2*d)