Integrand size = 20, antiderivative size = 186 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\frac {2}{105} a x \left (15 d+7 e x^2\right ) \sqrt {a-c x^4}+\frac {1}{63} x \left (9 d+7 e x^2\right ) \left (a-c x^4\right )^{3/2}+\frac {4 a^{11/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 c^{3/4} \sqrt {a-c x^4}}+\frac {4 a^{9/4} \left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{105 \sqrt [4]{c} \sqrt {a-c x^4}} \] Output:
2/105*a*x*(7*e*x^2+15*d)*(-c*x^4+a)^(1/2)+1/63*x*(7*e*x^2+9*d)*(-c*x^4+a)^ (3/2)+4/15*a^(11/4)*e*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^( 3/4)/(-c*x^4+a)^(1/2)+4/105*a^(9/4)*(15*d-7*a^(1/2)*e/c^(1/2))*(1-c*x^4/a) ^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(1/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.42 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\frac {a \sqrt {a-c x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)*(a - c*x^4)^(3/2),x]
Output:
(a*Sqrt[a - c*x^4]*(3*d*x*Hypergeometric2F1[-3/2, 1/4, 5/4, (c*x^4)/a] + e *x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[1 - (c*x^4)/a] )
Time = 0.68 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1491, 27, 1491, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {1}{21} \int 2 a \left (7 e x^2+9 d\right ) \sqrt {a-c x^4}dx+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} a \int \left (7 e x^2+9 d\right ) \sqrt {a-c x^4}dx+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {2}{21} a \left (\frac {1}{15} \int \frac {6 a \left (7 e x^2+15 d\right )}{\sqrt {a-c x^4}}dx+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \int \frac {7 e x^2+15 d}{\sqrt {a-c x^4}}dx+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {7 \sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {7 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {7 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\frac {7 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\frac {7 e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\frac {7 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{21} a \left (\frac {2}{5} a \left (\frac {7 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (15 d-\frac {7 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
Input:
Int[(d + e*x^2)*(a - c*x^4)^(3/2),x]
Output:
(x*(9*d + 7*e*x^2)*(a - c*x^4)^(3/2))/63 + (2*a*((x*(15*d + 7*e*x^2)*Sqrt[ a - c*x^4])/5 + (2*a*((7*a^(3/4)*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c ^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(15*d - (7*S qrt[a]*e)/Sqrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4 )], -1])/(c^(1/4)*Sqrt[a - c*x^4])))/5))/21
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*( d*(4*p + 3) + e*(4*p + 1)*x^2)*((a + c*x^4)^p/((4*p + 1)*(4*p + 3))), x] + Simp[2*(p/((4*p + 1)*(4*p + 3))) Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 1))*x^2, x]*(a + c*x^4)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c *d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Time = 1.09 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(\frac {x \left (-35 c e \,x^{6}-45 c d \,x^{4}+77 a e \,x^{2}+135 a d \right ) \sqrt {-c \,x^{4}+a}}{315}+\frac {4 a^{2} \left (\frac {15 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )}{105}\) | \(200\) |
| default | \(d \left (-\frac {c \,x^{5} \sqrt {-c \,x^{4}+a}}{7}+\frac {3 a x \sqrt {-c \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e \left (-\frac {c \,x^{7} \sqrt {-c \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {-c \,x^{4}+a}}{45}-\frac {4 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(224\) |
| elliptic | \(-\frac {c e \,x^{7} \sqrt {-c \,x^{4}+a}}{9}-\frac {c d \,x^{5} \sqrt {-c \,x^{4}+a}}{7}+\frac {11 a e \,x^{3} \sqrt {-c \,x^{4}+a}}{45}+\frac {3 a d x \sqrt {-c \,x^{4}+a}}{7}+\frac {4 a^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {4 a^{\frac {5}{2}} e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(224\) |
Input:
int((e*x^2+d)*(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/315*x*(-35*c*e*x^6-45*c*d*x^4+77*a*e*x^2+135*a*d)*(-c*x^4+a)^(1/2)+4/105 *a^2*(15*d/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/ a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2) )^(1/2),I)-7*e*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2) ^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF (x*(1/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)))
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.73 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=-\frac {84 \, a^{2} \sqrt {-c} e x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 12 \, {\left (15 \, a c d + 7 \, a^{2} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (35 \, c^{2} e x^{8} + 45 \, c^{2} d x^{6} - 77 \, a c e x^{4} - 135 \, a c d x^{2} + 84 \, a^{2} e\right )} \sqrt {-c x^{4} + a}}{315 \, c x} \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/315*(84*a^2*sqrt(-c)*e*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - 12*(15*a*c*d + 7*a^2*e)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/ c)^(1/4)/x), -1) + (35*c^2*e*x^8 + 45*c^2*d*x^6 - 77*a*c*e*x^4 - 135*a*c*d *x^2 + 84*a^2*e)*sqrt(-c*x^4 + a))/(c*x)
Time = 1.69 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {\sqrt {a} c d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {\sqrt {a} c e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)*(-c*x**4+a)**(3/2),x)
Output:
a**(3/2)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(2*I*pi )/a)/(4*gamma(5/4)) + a**(3/2)*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,) , c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4)) - sqrt(a)*c*d*x**5*gamma(5/4) *hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(9/4)) - s qrt(a)*c*e*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar(2* I*pi)/a)/(4*gamma(11/4))
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(3/2)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(e*x^2 + d), x)
Timed out. \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\int {\left (a-c\,x^4\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a - c*x^4)^(3/2)*(d + e*x^2),x)
Output:
int((a - c*x^4)^(3/2)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \, dx=\frac {3 \sqrt {-c \,x^{4}+a}\, a d x}{7}+\frac {11 \sqrt {-c \,x^{4}+a}\, a e \,x^{3}}{45}-\frac {\sqrt {-c \,x^{4}+a}\, c d \,x^{5}}{7}-\frac {\sqrt {-c \,x^{4}+a}\, c e \,x^{7}}{9}+\frac {4 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} d}{7}+\frac {4 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} e}{15} \] Input:
int((e*x^2+d)*(-c*x^4+a)^(3/2),x)
Output:
(135*sqrt(a - c*x**4)*a*d*x + 77*sqrt(a - c*x**4)*a*e*x**3 - 45*sqrt(a - c *x**4)*c*d*x**5 - 35*sqrt(a - c*x**4)*c*e*x**7 + 180*int(sqrt(a - c*x**4)/ (a - c*x**4),x)*a**2*d + 84*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a* *2*e)/315