Integrand size = 22, antiderivative size = 619 \[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 d \sqrt {d x} \left (8 b^4-21 a b^2 c-30 a^2 c^2+3 b c \left (8 b^2-31 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3}-\frac {2 d \sqrt {d x} \left (3 \left (2 b^2+a c\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2}+\frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {4 \sqrt {2} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) d^{3/2} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{1155 c^{9/2} \sqrt {a+x (b+c x)}}+\frac {\sqrt {2} \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \left (4 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )-\frac {a c \left (8 b^4-51 a b^2 c+60 a^2 c^2\right )}{b+\sqrt {b^2-4 a c}}\right ) d^{3/2} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{1155 c^{9/2} \sqrt {a+x (b+c x)}} \] Output:
2/1155*d*(d*x)^(1/2)*(8*b^4-21*a*b^2*c-30*a^2*c^2+3*b*c*(-31*a*c+8*b^2)*x) *(c*x^2+b*x+a)^(1/2)/c^3-2/231*d*(d*x)^(1/2)*(14*b*c*x+3*a*c+6*b^2)*(c*x^2 +b*x+a)^(3/2)/c^2+2/11*d*(d*x)^(1/2)*(c*x^2+b*x+a)^(5/2)/c-4/1155*2^(1/2)* b*(-9*a*c+2*b^2)*(-3*a*c+b^2)*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a*c+b^2 )^(1/2))*d^(3/2)*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a* c+b^2)^(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^2 )^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/ 2))/c^(9/2)/(a+x*(c*x+b))^(1/2)+1/1155*2^(1/2)*(-b+(-4*a*c+b^2)^(1/2))^(1/ 2)*(b+(-4*a*c+b^2)^(1/2))*(4*b*(-9*a*c+2*b^2)*(-3*a*c+b^2)-a*c*(60*a^2*c^2 -51*a*b^2*c+8*b^4)/(b+(-4*a*c+b^2)^(1/2)))*d^(3/2)*(1+2*c*x/(b-(-4*a*c+b^2 )^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(2^(1/2)*c ^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+b^2)^ (1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2))/c^(9/2)/(a+x*(c*x+b))^(1/2)
Result contains complex when optimal does not.
Time = 24.84 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.02 \[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {(d x)^{3/2} \left (-16 b \left (2 b^4-15 a b^2 c+27 a^2 c^2\right ) (a+x (b+c x))+2 c x (a+x (b+c x)) \left (8 b^4-6 b^3 c x+b^2 c \left (-51 a+5 c x^2\right )+4 b c^2 x \left (8 a+35 c x^2\right )+15 c^2 \left (4 a^2+13 a c x^2+7 c^2 x^4\right )\right )+\frac {4 i b \left (2 b^4-15 a b^2 c+27 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}-\frac {i \left (-8 b^6+68 a b^4 c-159 a^2 b^2 c^2+60 a^3 c^3+8 b^5 \sqrt {b^2-4 a c}-60 a b^3 c \sqrt {b^2-4 a c}+108 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{1155 c^4 x^2 \sqrt {a+x (b+c x)}} \] Input:
Integrate[(d*x)^(3/2)*(a + b*x + c*x^2)^(3/2),x]
Output:
((d*x)^(3/2)*(-16*b*(2*b^4 - 15*a*b^2*c + 27*a^2*c^2)*(a + x*(b + c*x)) + 2*c*x*(a + x*(b + c*x))*(8*b^4 - 6*b^3*c*x + b^2*c*(-51*a + 5*c*x^2) + 4*b *c^2*x*(8*a + 35*c*x^2) + 15*c^2*(4*a^2 + 13*a*c*x^2 + 7*c^2*x^4)) + ((4*I )*b*(2*b^4 - 15*a*b^2*c + 27*a^2*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4 *a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a* c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4 *a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] - (I*(-8*b^6 + 68*a*b^4*c - 159*a ^2*b^2*c^2 + 60*a^3*c^3 + 8*b^5*Sqrt[b^2 - 4*a*c] - 60*a*b^3*c*Sqrt[b^2 - 4*a*c] + 108*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] )/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]))/(1155*c^4*x^2*Sqrt[a + x*(b + c*x)])
Time = 1.41 (sec) , antiderivative size = 518, normalized size of antiderivative = 0.84, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1166, 27, 1231, 27, 1231, 27, 1241, 1240, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {2 \int -\frac {d^2 (a+6 b x) \left (c x^2+b x+a\right )^{3/2}}{2 \sqrt {d x}}dx}{11 c}+\frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \int \frac {(a+6 b x) \left (c x^2+b x+a\right )^{3/2}}{\sqrt {d x}}dx}{11 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {2 \int \frac {3 d^2 \left (2 a \left (b^2-3 a c\right )+b \left (8 b^2-31 a c\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d x}}dx}{21 c d^2}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\int \frac {\left (2 a \left (b^2-3 a c\right )+b \left (8 b^2-31 a c\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d x}}dx}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \int \frac {d^2 \left (a \left (8 b^4-51 a c b^2+60 a^2 c^2\right )+8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x\right )}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{15 c d^2}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {\int \frac {a \left (8 b^4-51 a c b^2+60 a^2 c^2\right )+8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{15 c}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 1241 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {\sqrt {x} \int \frac {a \left (8 b^4-51 a c b^2+60 a^2 c^2\right )+8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{15 c \sqrt {d x}}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \int \frac {a \left (8 b^4-51 a c b^2+60 a^2 c^2\right )+8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{15 c \sqrt {d x}}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {8 \sqrt {a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\sqrt {a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{7 c}\right )}{11 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 d \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {d^2 \left (\frac {2 \sqrt {d x} \left (3 \left (a c+2 b^2\right )+14 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 c d}-\frac {\frac {2 \sqrt {d x} \left (-30 a^2 c^2+3 b c x \left (8 b^2-31 a c\right )-21 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{7 c}\right )}{11 c}\) |
Input:
Int[(d*x)^(3/2)*(a + b*x + c*x^2)^(3/2),x]
Output:
(2*d*Sqrt[d*x]*(a + b*x + c*x^2)^(5/2))/(11*c) - (d^2*((2*Sqrt[d*x]*(3*(2* b^2 + a*c) + 14*b*c*x)*(a + b*x + c*x^2)^(3/2))/(21*c*d) - ((2*Sqrt[d*x]*( 8*b^4 - 21*a*b^2*c - 30*a^2*c^2 + 3*b*c*(8*b^2 - 31*a*c)*x)*Sqrt[a + b*x + c*x^2])/(15*c*d) - (2*Sqrt[x]*((-8*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*(-((Sq rt[x]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*Arc Tan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqr t[a + b*x + c*x^2])))/Sqrt[c] + (a^(1/4)*((8*b*(2*b^2 - 9*a*c)*(b^2 - 3*a* c))/Sqrt[c] + Sqrt[a]*(8*b^4 - 51*a*b^2*c + 60*a^2*c^2))*(Sqrt[a] + Sqrt[c ]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c ^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x + c*x^2])))/(15*c*Sqrt[d*x]))/(7*c)))/(11*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ )^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(1254\) vs. \(2(519)=1038\).
Time = 2.81 (sec) , antiderivative size = 1255, normalized size of antiderivative = 2.03
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1255\) |
| risch | \(\text {Expression too large to display}\) | \(1376\) |
| default | \(\text {Expression too large to display}\) | \(2278\) |
Input:
int((d*x)^(3/2)*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/d/x*(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)*(d*x*(c*x^2+b*x+a))^(1/2)*(2/11*c*d* x^4*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)+8/33*b*d*x^3*(c*d*x^3+b*d*x^2+a*d*x)^(1/ 2)+2/7*((2*a*c+b^2)*d^2-9/11*a*d^2*c-32/33*b^2*d^2)/c/d*x^2*(c*d*x^3+b*d*x ^2+a*d*x)^(1/2)+2/5*(38/33*a*b*d^2-6/7*((2*a*c+b^2)*d^2-9/11*a*d^2*c-32/33 *b^2*d^2)/c*b)/c/d*x*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)+2/3*(a^2*d^2-5/7*((2*a* c+b^2)*d^2-9/11*a*d^2*c-32/33*b^2*d^2)/c*a-4/5*(38/33*a*b*d^2-6/7*((2*a*c+ b^2)*d^2-9/11*a*d^2*c-32/33*b^2*d^2)/c*b)/c*b)/c/d*(c*d*x^3+b*d*x^2+a*d*x) ^(1/2)-1/3*(a^2*d^2-5/7*((2*a*c+b^2)*d^2-9/11*a*d^2*c-32/33*b^2*d^2)/c*a-4 /5*(38/33*a*b*d^2-6/7*((2*a*c+b^2)*d^2-9/11*a*d^2*c-32/33*b^2*d^2)/c*b)/c* b)/c^2*a*(b+(-4*a*c+b^2)^(1/2))*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/ (b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*( b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+( -4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*EllipticF(2^(1/2)* ((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2* (b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+ b^2)^(1/2))))^(1/2))+(-3/5*(38/33*a*b*d^2-6/7*((2*a*c+b^2)*d^2-9/11*a*d^2* c-32/33*b^2*d^2)/c*b)/c*a-2/3*(a^2*d^2-5/7*((2*a*c+b^2)*d^2-9/11*a*d^2*c-3 2/33*b^2*d^2)/c*a-4/5*(38/33*a*b*d^2-6/7*((2*a*c+b^2)*d^2-9/11*a*d^2*c-32/ 33*b^2*d^2)/c*b)/c*b)/c*b)*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4 *a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a...
Time = 0.11 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.48 \[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (16 \, b^{6} - 144 \, a b^{4} c + 369 \, a^{2} b^{2} c^{2} - 180 \, a^{3} c^{3}\right )} \sqrt {c d} d {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 24 \, {\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 27 \, a^{2} b c^{3}\right )} \sqrt {c d} d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (105 \, c^{6} d x^{4} + 140 \, b c^{5} d x^{3} + 5 \, {\left (b^{2} c^{4} + 39 \, a c^{5}\right )} d x^{2} - 2 \, {\left (3 \, b^{3} c^{3} - 16 \, a b c^{4}\right )} d x + {\left (8 \, b^{4} c^{2} - 51 \, a b^{2} c^{3} + 60 \, a^{2} c^{4}\right )} d\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{3465 \, c^{5}} \] Input:
integrate((d*x)^(3/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
2/3465*((16*b^6 - 144*a*b^4*c + 369*a^2*b^2*c^2 - 180*a^3*c^3)*sqrt(c*d)*d *weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1 /3*(3*c*x + b)/c) + 24*(2*b^5*c - 15*a*b^3*c^2 + 27*a^2*b*c^3)*sqrt(c*d)*d *weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, weier strassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3* c*x + b)/c)) + 3*(105*c^6*d*x^4 + 140*b*c^5*d*x^3 + 5*(b^2*c^4 + 39*a*c^5) *d*x^2 - 2*(3*b^3*c^3 - 16*a*b*c^4)*d*x + (8*b^4*c^2 - 51*a*b^2*c^3 + 60*a ^2*c^4)*d)*sqrt(c*x^2 + b*x + a)*sqrt(d*x))/c^5
\[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \left (d x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((d*x)**(3/2)*(c*x**2+b*x+a)**(3/2),x)
Output:
Integral((d*x)**(3/2)*(a + b*x + c*x**2)**(3/2), x)
\[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \left (d x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x)^(3/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(3/2)*(d*x)^(3/2), x)
\[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \left (d x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x)^(3/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(3/2)*(d*x)^(3/2), x)
Timed out. \[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (d\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \] Input:
int((d*x)^(3/2)*(a + b*x + c*x^2)^(3/2),x)
Output:
int((d*x)^(3/2)*(a + b*x + c*x^2)^(3/2), x)
\[ \int (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {d}\, d \left (-96 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a^{2} c +18 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}+64 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a b c x +390 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} x^{2}-12 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b^{3} x +10 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,x^{2}+280 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} x^{3}+210 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, c^{3} x^{4}+324 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a^{2} c^{2}-180 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a \,b^{2} c +24 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) b^{4}+48 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{3} c -9 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{2} b^{2}\right )}{1155 c^{2}} \] Input:
int((d*x)^(3/2)*(c*x^2+b*x+a)^(3/2),x)
Output:
(sqrt(d)*d*( - 96*sqrt(x)*sqrt(a + b*x + c*x**2)*a**2*c + 18*sqrt(x)*sqrt( a + b*x + c*x**2)*a*b**2 + 64*sqrt(x)*sqrt(a + b*x + c*x**2)*a*b*c*x + 390 *sqrt(x)*sqrt(a + b*x + c*x**2)*a*c**2*x**2 - 12*sqrt(x)*sqrt(a + b*x + c* x**2)*b**3*x + 10*sqrt(x)*sqrt(a + b*x + c*x**2)*b**2*c*x**2 + 280*sqrt(x) *sqrt(a + b*x + c*x**2)*b*c**2*x**3 + 210*sqrt(x)*sqrt(a + b*x + c*x**2)*c **3*x**4 + 324*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x )*a**2*c**2 - 180*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2 ),x)*a*b**2*c + 24*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x** 2),x)*b**4 + 48*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x** 3),x)*a**3*c - 9*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x* *3),x)*a**2*b**2))/(1155*c**2)