Integrand size = 28, antiderivative size = 447 \[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=-\frac {8 b^4 (11 b B-21 A c) x^{3/2} \left (b+c x^2\right )}{3315 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {8 b^3 (11 b B-21 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{9945 c^3}-\frac {8 b^2 (11 b B-21 A c) x^{5/2} \sqrt {b x^2+c x^4}}{13923 c^2}-\frac {4 b (11 b B-21 A c) x^{9/2} \sqrt {b x^2+c x^4}}{1547 c}-\frac {2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}+\frac {8 b^{17/4} (11 b B-21 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {4 b^{17/4} (11 b B-21 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3315 c^{15/4} \sqrt {b x^2+c x^4}} \] Output:
-8/3315*b^4*(-21*A*c+11*B*b)*x^(3/2)*(c*x^2+b)/c^(7/2)/(b^(1/2)+c^(1/2)*x) /(c*x^4+b*x^2)^(1/2)+8/9945*b^3*(-21*A*c+11*B*b)*x^(1/2)*(c*x^4+b*x^2)^(1/ 2)/c^3-8/13923*b^2*(-21*A*c+11*B*b)*x^(5/2)*(c*x^4+b*x^2)^(1/2)/c^2-4/1547 *b*(-21*A*c+11*B*b)*x^(9/2)*(c*x^4+b*x^2)^(1/2)/c-2/357*(-21*A*c+11*B*b)*x ^(5/2)*(c*x^4+b*x^2)^(3/2)/c+2/21*B*x^(1/2)*(c*x^4+b*x^2)^(5/2)/c+8/3315*b ^(17/4)*(-21*A*c+11*B*b)*x*(b^(1/2)+c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c^(1/2) *x)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x^(1/2)/b^(1/4))),1/2*2^(1/2)) /c^(15/4)/(c*x^4+b*x^2)^(1/2)-4/3315*b^(17/4)*(-21*A*c+11*B*b)*x*(b^(1/2)+ c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arcta n(c^(1/4)*x^(1/2)/b^(1/4)),1/2*2^(1/2))/c^(15/4)/(c*x^4+b*x^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.31 \[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left (\left (b+c x^2\right )^2 \sqrt {1+\frac {c x^2}{b}} \left (77 b^2 B+13 c^2 x^2 \left (21 A+17 B x^2\right )-b c \left (147 A+143 B x^2\right )\right )+7 b^3 (-11 b B+21 A c) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{b}\right )\right )}{4641 c^3 \sqrt {1+\frac {c x^2}{b}}} \] Input:
Integrate[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]
Output:
(2*Sqrt[x]*Sqrt[x^2*(b + c*x^2)]*((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(77*b^ 2*B + 13*c^2*x^2*(21*A + 17*B*x^2) - b*c*(147*A + 143*B*x^2)) + 7*b^3*(-11 *b*B + 21*A*c)*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2)/b)]))/(4641*c^3 *Sqrt[1 + (c*x^2)/b])
Time = 1.05 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1945, 1426, 1426, 1429, 1429, 1431, 266, 834, 27, 761, 1510}
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.
\(\displaystyle \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1945 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \int x^{3/2} \left (c x^4+b x^2\right )^{3/2}dx}{21 c}\) |
\(\Big \downarrow \) 1426 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \int x^{7/2} \sqrt {c x^4+b x^2}dx+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 1426 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \int \frac {x^{11/2}}{\sqrt {c x^4+b x^2}}dx+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 1429 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \int \frac {x^{7/2}}{\sqrt {c x^4+b x^2}}dx}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 1429 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {3 b \int \frac {x^{3/2}}{\sqrt {c x^4+b x^2}}dx}{5 c}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 1431 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {3 b x \sqrt {b+c x^2} \int \frac {\sqrt {x}}{\sqrt {c x^2+b}}dx}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \int \frac {x}{\sqrt {c x^2+b}}d\sqrt {x}}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {b} \sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {2 B \sqrt {x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac {(11 b B-21 A c) \left (\frac {6}{17} b \left (\frac {2}{13} b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {b+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x^2}}{\sqrt {b}+\sqrt {c} x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )+\frac {2}{13} x^{9/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}\right )}{21 c}\) |
Input:
Int[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]
Output:
(2*B*Sqrt[x]*(b*x^2 + c*x^4)^(5/2))/(21*c) - ((11*b*B - 21*A*c)*((2*x^(5/2 )*(b*x^2 + c*x^4)^(3/2))/17 + (6*b*((2*x^(9/2)*Sqrt[b*x^2 + c*x^4])/13 + ( 2*b*((2*x^(5/2)*Sqrt[b*x^2 + c*x^4])/(9*c) - (7*b*((2*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(5*c) - (6*b*x*Sqrt[b + c*x^2]*(-((-((Sqrt[x]*Sqrt[b + c*x^2])/(S qrt[b] + Sqrt[c]*x)) + (b^(1/4)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sq rt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2]) /(c^(1/4)*Sqrt[b + c*x^2]))/Sqrt[c]) + (b^(1/4)*(Sqrt[b] + Sqrt[c]*x)*Sqrt [(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x]) /b^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[b + c*x^2])))/(5*c*Sqrt[b*x^2 + c*x^4]))) /(9*c)))/13))/17))/(21*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp [(d*x)^(m + 1)*((b*x^2 + c*x^4)^p/(d*(m + 4*p + 1))), x] + Simp[2*b*(p/(d^2 *(m + 4*p + 1))) Int[(d*x)^(m + 2)*(b*x^2 + c*x^4)^(p - 1), x], x] /; Fre eQ[{b, c, d, m, p}, x] && !IntegerQ[p] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0]
Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp [d^3*(d*x)^(m - 3)*((b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[b *d^2*((m + 2*p - 1)/(c*(m + 4*p + 1))) Int[(d*x)^(m - 2)*(b*x^2 + c*x^4)^ p, x], x] /; FreeQ[{b, c, d, m, p}, x] && !IntegerQ[p] && GtQ[m + 2*p - 1, 0] && NeQ[m + 4*p + 1, 0]
Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp [(b*x^2 + c*x^4)^p/((d*x)^(2*p)*(b + c*x^2)^p) Int[(d*x)^(m + 2*p)*(b + c *x^2)^p, x], x] /; FreeQ[{b, c, d, m, p}, x] && !IntegerQ[p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)) Int[(e* x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p *(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
Time = 1.61 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {2 \sqrt {x}\, \left (-3315 B \,x^{8} c^{4}-4095 A \,x^{6} c^{4}-4485 B \,x^{6} b \,c^{3}-5985 A \,x^{4} b \,c^{3}-180 B \,x^{4} b^{2} c^{2}-420 A \,b^{2} c^{2} x^{2}+220 B \,b^{3} c \,x^{2}+588 A \,b^{3} c -308 B \,b^{4}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{69615 c^{3}}+\frac {4 b^{4} \left (21 A c -11 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{3315 c^{4} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(315\) |
default | \(\frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3315 B \,c^{6} x^{12}+4095 A \,c^{6} x^{10}+7800 B b \,c^{5} x^{10}+10080 A b \,c^{5} x^{8}+4665 B \,b^{2} c^{4} x^{8}+6405 A \,b^{2} c^{4} x^{6}-40 B \,b^{3} c^{3} x^{6}+1764 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{5} c -882 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{5} c -924 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{6}+462 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{6}-168 A \,b^{3} c^{3} x^{4}+88 B \,b^{4} c^{2} x^{4}-588 A \,b^{4} c^{2} x^{2}+308 B \,b^{5} c \,x^{2}\right )}{69615 x^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{2} c^{4}}\) | \(494\) |
Input:
int(x^(3/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/69615/c^3*x^(1/2)*(-3315*B*c^4*x^8-4095*A*c^4*x^6-4485*B*b*c^3*x^6-5985 *A*b*c^3*x^4-180*B*b^2*c^2*x^4-420*A*b^2*c^2*x^2+220*B*b^3*c*x^2+588*A*b^3 *c-308*B*b^4)*(x^2*(c*x^2+b))^(1/2)+4/3315*b^4/c^4*(21*A*c-11*B*b)*(-b*c)^ (1/2)*((x+1/c*(-b*c)^(1/2))*c/(-b*c)^(1/2))^(1/2)*(-2*(x-1/c*(-b*c)^(1/2)) *c/(-b*c)^(1/2))^(1/2)*(-c/(-b*c)^(1/2)*x)^(1/2)/(c*x^3+b*x)^(1/2)*(-2/c*( -b*c)^(1/2)*EllipticE(((x+1/c*(-b*c)^(1/2))*c/(-b*c)^(1/2))^(1/2),1/2*2^(1 /2))+1/c*(-b*c)^(1/2)*EllipticF(((x+1/c*(-b*c)^(1/2))*c/(-b*c)^(1/2))^(1/2 ),1/2*2^(1/2)))*(x^2*(c*x^2+b))^(1/2)/x^(3/2)/(c*x^2+b)*(x*(c*x^2+b))^(1/2 )
Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.34 \[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \, {\left (84 \, {\left (11 \, B b^{5} - 21 \, A b^{4} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (3315 \, B c^{5} x^{8} + 195 \, {\left (23 \, B b c^{4} + 21 \, A c^{5}\right )} x^{6} + 308 \, B b^{4} c - 588 \, A b^{3} c^{2} + 45 \, {\left (4 \, B b^{2} c^{3} + 133 \, A b c^{4}\right )} x^{4} - 20 \, {\left (11 \, B b^{3} c^{2} - 21 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{69615 \, c^{4}} \] Input:
integrate(x^(3/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
Output:
2/69615*(84*(11*B*b^5 - 21*A*b^4*c)*sqrt(c)*weierstrassZeta(-4*b/c, 0, wei erstrassPInverse(-4*b/c, 0, x)) + (3315*B*c^5*x^8 + 195*(23*B*b*c^4 + 21*A *c^5)*x^6 + 308*B*b^4*c - 588*A*b^3*c^2 + 45*(4*B*b^2*c^3 + 133*A*b*c^4)*x ^4 - 20*(11*B*b^3*c^2 - 21*A*b^2*c^3)*x^2)*sqrt(c*x^4 + b*x^2)*sqrt(x))/c^ 4
Timed out. \[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate(x**(3/2)*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)
Output:
Timed out
\[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} x^{\frac {3}{2}} \,d x } \] Input:
integrate(x^(3/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(3/2), x)
\[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} x^{\frac {3}{2}} \,d x } \] Input:
integrate(x^(3/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(3/2), x)
Timed out. \[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int x^{3/2}\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \] Input:
int(x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x)
\[ \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {-\frac {56 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,b^{3} c x}{3315}+\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,b^{2} c^{2} x^{3}}{663}+\frac {38 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a b \,c^{3} x^{5}}{221}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,c^{4} x^{7}}{17}+\frac {88 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{5} x}{9945}-\frac {88 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{4} c \,x^{3}}{13923}+\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{3} c^{2} x^{5}}{1547}+\frac {46 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{2} c^{3} x^{7}}{357}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b \,c^{4} x^{9}}{21}+\frac {28 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{2}+b}d x \right ) a \,b^{4} c}{1105}-\frac {44 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{2}+b}d x \right ) b^{6}}{3315}}{c^{3}} \] Input:
int(x^(3/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)
Output:
(2*( - 588*sqrt(x)*sqrt(b + c*x**2)*a*b**3*c*x + 420*sqrt(x)*sqrt(b + c*x* *2)*a*b**2*c**2*x**3 + 5985*sqrt(x)*sqrt(b + c*x**2)*a*b*c**3*x**5 + 4095* sqrt(x)*sqrt(b + c*x**2)*a*c**4*x**7 + 308*sqrt(x)*sqrt(b + c*x**2)*b**5*x - 220*sqrt(x)*sqrt(b + c*x**2)*b**4*c*x**3 + 180*sqrt(x)*sqrt(b + c*x**2) *b**3*c**2*x**5 + 4485*sqrt(x)*sqrt(b + c*x**2)*b**2*c**3*x**7 + 3315*sqrt (x)*sqrt(b + c*x**2)*b*c**4*x**9 + 882*int((sqrt(x)*sqrt(b + c*x**2))/(b + c*x**2),x)*a*b**4*c - 462*int((sqrt(x)*sqrt(b + c*x**2))/(b + c*x**2),x)* b**6))/(69615*c**3)