Integrand size = 28, antiderivative size = 486 \[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {88 b^5 (3 b B-5 A c) x^{3/2} \left (b+c x^2\right )}{16575 c^{9/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {88 b^4 (3 b B-5 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{49725 c^4}+\frac {88 b^3 (3 b B-5 A c) x^{5/2} \sqrt {b x^2+c x^4}}{69615 c^3}-\frac {8 b^2 (3 b B-5 A c) x^{9/2} \sqrt {b x^2+c x^4}}{7735 c^2}-\frac {4 b (3 b B-5 A c) x^{13/2} \sqrt {b x^2+c x^4}}{595 c}-\frac {2 (3 b B-5 A c) x^{9/2} \left (b x^2+c x^4\right )^{3/2}}{105 c}+\frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {88 b^{21/4} (3 b B-5 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{16575 c^{19/4} \sqrt {b x^2+c x^4}}+\frac {44 b^{21/4} (3 b B-5 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{16575 c^{19/4} \sqrt {b x^2+c x^4}} \] Output:
88/16575*b^5*(-5*A*c+3*B*b)*x^(3/2)*(c*x^2+b)/c^(9/2)/(b^(1/2)+c^(1/2)*x)/ (c*x^4+b*x^2)^(1/2)-88/49725*b^4*(-5*A*c+3*B*b)*x^(1/2)*(c*x^4+b*x^2)^(1/2 )/c^4+88/69615*b^3*(-5*A*c+3*B*b)*x^(5/2)*(c*x^4+b*x^2)^(1/2)/c^3-8/7735*b ^2*(-5*A*c+3*B*b)*x^(9/2)*(c*x^4+b*x^2)^(1/2)/c^2-4/595*b*(-5*A*c+3*B*b)*x ^(13/2)*(c*x^4+b*x^2)^(1/2)/c-2/105*(-5*A*c+3*B*b)*x^(9/2)*(c*x^4+b*x^2)^( 3/2)/c+2/25*B*x^(5/2)*(c*x^4+b*x^2)^(5/2)/c-88/16575*b^(21/4)*(-5*A*c+3*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^(19/4)/(c*x^4+b*x^2 )^(1/2)+44/16575*b^(21/4)*(-5*A*c+3*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*arctan(c^(1/4)*x^(1/2)/b^(1 /4)),1/2*2^(1/2))/c^(19/4)/(c*x^4+b*x^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.18 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.33 \[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left (-\left (b+c x^2\right )^2 \sqrt {1+\frac {c x^2}{b}} \left (1155 b^3 B-221 c^3 x^4 \left (25 A+21 B x^2\right )-55 b^2 c \left (35 A+39 B x^2\right )+65 b c^2 x^2 \left (55 A+51 B x^2\right )\right )+385 b^4 (3 b B-5 A c) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{b}\right )\right )}{116025 c^4 \sqrt {1+\frac {c x^2}{b}}} \] Input:
Integrate[x^(7/2)*(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]*(115 5*b^3*B - 221*c^3*x^4*(25*A + 21*B*x^2) - 55*b^2*c*(35*A + 39*B*x^2) + 65* b*c^2*x^2*(55*A + 51*B*x^2))) + 385*b^4*(3*b*B - 5*A*c)*Hypergeometric2F1[ -3/2, 3/4, 7/4, -((c*x^2)/b)]))/(116025*c^4*Sqrt[1 + (c*x^2)/b])
Time = 1.08 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1945, 1426, 1426, 1429, 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.
| \(\displaystyle \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1945 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \int x^{7/2} \left (c x^4+b x^2\right )^{3/2}dx}{5 c}\) |
| \(\Big \downarrow \) 1426 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \int x^{11/2} \sqrt {c x^4+b x^2}dx+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1426 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \int \frac {x^{15/2}}{\sqrt {c x^4+b x^2}}dx+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1429 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \int \frac {x^{11/2}}{\sqrt {c x^4+b x^2}}dx}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1429 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \int \frac {x^{7/2}}{\sqrt {c x^4+b x^2}}dx}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1429 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {3 b \int \frac {x^{3/2}}{\sqrt {c x^4+b x^2}}dx}{5 c}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1431 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {3 b x \sqrt {b+c x^2} \int \frac {\sqrt {x}}{\sqrt {c x^2+b}}dx}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \int \frac {x}{\sqrt {c x^2+b}}d\sqrt {x}}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {b} \sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B x^{5/2} \left (b x^2+c x^4\right )^{5/2}}{25 c}-\frac {(3 b B-5 A c) \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {11 b \left (\frac {2 x^{5/2} \sqrt {b x^2+c x^4}}{9 c}-\frac {7 b \left (\frac {2 \sqrt {x} \sqrt {b x^2+c x^4}}{5 c}-\frac {6 b x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {b+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x^2}}{\sqrt {b}+\sqrt {c} x}}{\sqrt {c}}\right )}{5 c \sqrt {b x^2+c x^4}}\right )}{9 c}\right )}{13 c}\right )+\frac {2}{17} x^{13/2} \sqrt {b x^2+c x^4}\right )+\frac {2}{21} x^{9/2} \left (b x^2+c x^4\right )^{3/2}\right )}{5 c}\) |
Input:
Int[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]
Output:
(2*B*x^(5/2)*(b*x^2 + c*x^4)^(5/2))/(25*c) - ((3*b*B - 5*A*c)*((2*x^(9/2)* (b*x^2 + c*x^4)^(3/2))/21 + (2*b*((2*x^(13/2)*Sqrt[b*x^2 + c*x^4])/17 + (2 *b*((2*x^(9/2)*Sqrt[b*x^2 + c*x^4])/(13*c) - (11*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*Sqr t[b + c*x^2]*(-((-((Sqrt[x]*Sqrt[b + c*x^2])/(Sqrt[b] + Sqrt[c]*x)) + (b^( 1/4)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip ticE[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] + Sqr t[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*c)))/17))/7))/ (5*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 = 2.18 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {2 \sqrt {x}\, \left (13923 B \,c^{5} x^{10}+16575 A \,c^{5} x^{8}+17901 B b \,c^{4} x^{8}+22425 A b \,c^{4} x^{6}+468 B \,b^{2} c^{3} x^{6}+900 A \,b^{2} c^{3} x^{4}-540 B \,b^{3} c^{2} x^{4}-1100 A \,b^{3} c^{2} x^{2}+660 B \,b^{4} c \,x^{2}+1540 A \,b^{4} c -924 b^{5} B \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{348075 c^{4}}-\frac {44 b^{5} \left (5 A c -3 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{16575 c^{5} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(339\) |
| default | \(-\frac {2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (-13923 B \,c^{7} x^{14}-16575 A \,c^{7} x^{12}-31824 B b \,c^{6} x^{12}-39000 A b \,c^{6} x^{10}-18369 B \,b^{2} c^{5} x^{10}-23325 A \,b^{2} c^{5} x^{8}+72 B \,b^{3} c^{4} x^{8}+200 A \,b^{3} c^{4} x^{6}-120 B \,b^{4} c^{3} x^{6}+4620 A \,b^{6} c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-2310 A \,b^{6} c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-2772 B \,b^{7} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+1386 B \,b^{7} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-440 A \,b^{4} c^{3} x^{4}+264 B \,b^{5} c^{2} x^{4}-1540 A \,b^{5} c^{2} x^{2}+924 B \,b^{6} c \,x^{2}\right )}{348075 x^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{2} c^{5}}\) | \(518\) |
Input:
int(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/348075/c^4*x^(1/2)*(13923*B*c^5*x^10+16575*A*c^5*x^8+17901*B*b*c^4*x^8+2 2425*A*b*c^4*x^6+468*B*b^2*c^3*x^6+900*A*b^2*c^3*x^4-540*B*b^3*c^2*x^4-110 0*A*b^3*c^2*x^2+660*B*b^4*c*x^2+1540*A*b^4*c-924*B*b^5)*(x^2*(c*x^2+b))^(1 /2)-44/16575*b^5/c^5*(5*A*c-3*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 = 0.12 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.36 \[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=-\frac {2 \, {\left (924 \, {\left (3 \, B b^{6} - 5 \, A b^{5} c\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (13923 \, B c^{6} x^{10} + 663 \, {\left (27 \, B b c^{5} + 25 \, A c^{6}\right )} x^{8} - 924 \, B b^{5} c + 1540 \, A b^{4} c^{2} + 39 \, {\left (12 \, B b^{2} c^{4} + 575 \, A b c^{5}\right )} x^{6} - 180 \, {\left (3 \, B b^{3} c^{3} - 5 \, A b^{2} c^{4}\right )} x^{4} + 220 \, {\left (3 \, B b^{4} c^{2} - 5 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{348075 \, c^{5}} \] Input:
integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
Output:
-2/348075*(924*(3*B*b^6 - 5*A*b^5*c)*sqrt(c)*weierstrassZeta(-4*b/c, 0, we ierstrassPInverse(-4*b/c, 0, x)) - (13923*B*c^6*x^10 + 663*(27*B*b*c^5 + 2 5*A*c^6)*x^8 - 924*B*b^5*c + 1540*A*b^4*c^2 + 39*(12*B*b^2*c^4 + 575*A*b*c ^5)*x^6 - 180*(3*B*b^3*c^3 - 5*A*b^2*c^4)*x^4 + 220*(3*B*b^4*c^2 - 5*A*b^3 *c^3)*x^2)*sqrt(c*x^4 + b*x^2)*sqrt(x))/c^5
Timed out. \[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate(x**(7/2)*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)
Output:
Timed out
\[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} x^{\frac {7}{2}} \,d x } \] Input:
integrate(x^(7/2)*(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^(7/2), x)
\[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} x^{\frac {7}{2}} \,d x } \] Input:
integrate(x^(7/2)*(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^(7/2), x)
Timed out. \[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int x^{7/2}\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \] Input:
int(x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x)
\[ \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {\frac {88 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,b^{4} c x}{9945}-\frac {88 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,b^{3} c^{2} x^{3}}{13923}+\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,b^{2} c^{3} x^{5}}{1547}+\frac {46 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a b \,c^{4} x^{7}}{357}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, a \,c^{5} x^{9}}{21}-\frac {88 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{6} x}{16575}+\frac {88 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{5} c \,x^{3}}{23205}-\frac {24 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{4} c^{2} x^{5}}{7735}+\frac {8 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{3} c^{3} x^{7}}{2975}+\frac {18 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b^{2} c^{4} x^{9}}{175}+\frac {2 \sqrt {x}\, \sqrt {c \,x^{2}+b}\, b \,c^{5} x^{11}}{25}-\frac {44 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{2}+b}d x \right ) a \,b^{5} c}{3315}+\frac {44 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c \,x^{2}+b}d x \right ) b^{7}}{5525}}{c^{4}} \] Input:
int(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)
Output:
(2*(1540*sqrt(x)*sqrt(b + c*x**2)*a*b**4*c*x - 1100*sqrt(x)*sqrt(b + c*x** 2)*a*b**3*c**2*x**3 + 900*sqrt(x)*sqrt(b + c*x**2)*a*b**2*c**3*x**5 + 2242 5*sqrt(x)*sqrt(b + c*x**2)*a*b*c**4*x**7 + 16575*sqrt(x)*sqrt(b + c*x**2)* a*c**5*x**9 - 924*sqrt(x)*sqrt(b + c*x**2)*b**6*x + 660*sqrt(x)*sqrt(b + c *x**2)*b**5*c*x**3 - 540*sqrt(x)*sqrt(b + c*x**2)*b**4*c**2*x**5 + 468*sqr t(x)*sqrt(b + c*x**2)*b**3*c**3*x**7 + 17901*sqrt(x)*sqrt(b + c*x**2)*b**2 *c**4*x**9 + 13923*sqrt(x)*sqrt(b + c*x**2)*b*c**5*x**11 - 2310*int((sqrt( x)*sqrt(b + c*x**2))/(b + c*x**2),x)*a*b**5*c + 1386*int((sqrt(x)*sqrt(b + c*x**2))/(b + c*x**2),x)*b**7))/(348075*c**4)