Integrand size = 19, antiderivative size = 329 \[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c}-\frac {8 a^3 c^2 \sqrt {c x} \sqrt {a+b x^2}}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac {8 a^{13/4} c^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^2}}-\frac {4 a^{13/4} c^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^2}} \]
2/13*(c*x)^(7/2)*(b*x^2+a)^(3/2)/c+8/195*a^2*c*(c*x)^(3/2)*(b*x^2+a)^(1/2) /b+4/39*a*(c*x)^(7/2)*(b*x^2+a)^(1/2)/c-8/65*a^3*c^2*(c*x)^(1/2)*(b*x^2+a) ^(1/2)/b^(3/2)/(a^(1/2)+x*b^(1/2))+8/65*a^(13/4)*c^(5/2)*(cos(2*arctan(b^( 1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)^(1/ 2)/a^(1/4)/c^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^ (1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2) ^(1/2)/b^(7/4)/(b*x^2+a)^(1/2)-4/65*a^(13/4)*c^(5/2)*(cos(2*arctan(b^(1/4) *(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a ^(1/4)/c^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2 ))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/ 2)/b^(7/4)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.27 \[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {2 c (c x)^{3/2} \sqrt {a+b x^2} \left (\left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{13 b \sqrt {1+\frac {b x^2}{a}}} \]
(2*c*(c*x)^(3/2)*Sqrt[a + b*x^2]*((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] - a^2* Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)]))/(13*b*Sqrt[1 + (b*x^2)/a ])
Time = 0.40 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {248, 248, 262, 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.
| \(\displaystyle \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {6}{13} a \int (c x)^{5/2} \sqrt {b x^2+a}dx+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \int \frac {(c x)^{5/2}}{\sqrt {b x^2+a}}dx+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 c (c x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 a c^2 \int \frac {\sqrt {c x}}{\sqrt {b x^2+a}}dx}{5 b}\right )+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 c (c x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {6 a c \int \frac {c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{5 b}\right )+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 c (c x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {6 a c \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\sqrt {a} c \int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {a} c \sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{5 b}\right )+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 c (c x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {6 a c \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{5 b}\right )+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 c (c x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {6 a c \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{5 b}\right )+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 c (c x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {6 a c \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {c^2 \sqrt {c x} \sqrt {a+b x^2}}{\sqrt {a} c+\sqrt {b} c x}}{\sqrt {b}}\right )}{5 b}\right )+\frac {2 (c x)^{7/2} \sqrt {a+b x^2}}{9 c}\right )+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}\) |
(2*(c*x)^(7/2)*(a + b*x^2)^(3/2))/(13*c) + (6*a*((2*(c*x)^(7/2)*Sqrt[a + b *x^2])/(9*c) + (2*a*((2*c*(c*x)^(3/2)*Sqrt[a + b*x^2])/(5*b) - (6*a*c*(-(( -((c^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(Sqrt[a]*c + Sqrt[b]*c*x)) + (a^(1/4)*Sq rt[c]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt [b]*c*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2 ])/(b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b]) + (a^(1/4)*Sqrt[c]*(Sqrt[a]*c + Sqr t[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x)^2]*EllipticF[ 2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^2])))/(5*b)))/9))/13
3.6.98.3.1 Defintions of rubi rules used
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] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, 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_) + (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]
Time = 2.02 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(-\frac {2 c^{2} \sqrt {c x}\, \left (-15 x^{8} b^{4}-40 a \,b^{3} x^{6}+12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{4}-6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{4}-29 a^{2} x^{4} b^{2}-4 a^{3} b \,x^{2}\right )}{195 x \sqrt {b \,x^{2}+a}\, b^{2}}\) | \(232\) |
| risch | \(\frac {2 x^{2} \left (15 b^{2} x^{4}+25 a b \,x^{2}+4 a^{2}\right ) \sqrt {b \,x^{2}+a}\, c^{3}}{195 b \sqrt {c x}}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) c^{3} \sqrt {c x \left (b \,x^{2}+a \right )}}{65 b^{2} \sqrt {b c \,x^{3}+a c x}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(240\) |
| elliptic | \(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {2 b \,c^{2} x^{5} \sqrt {b c \,x^{3}+a c x}}{13}+\frac {10 a \,c^{2} x^{3} \sqrt {b c \,x^{3}+a c x}}{39}+\frac {8 a^{2} c^{2} x \sqrt {b c \,x^{3}+a c x}}{195 b}-\frac {4 a^{3} c^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b c \,x^{3}+a c x}}\right )}{c x \sqrt {b \,x^{2}+a}}\) | \(270\) |
-2/195*c^2/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)/b^2*(-15*x^8*b^4-40*a*b^3*x^6+12* ((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b )^(1/2))^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a *b)^(1/2))^(1/2),1/2*2^(1/2))*a^4-6*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2 )*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/ 2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^4-29*a ^2*x^4*b^2-4*a^3*b*x^2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.26 \[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {2 \, {\left (12 \, \sqrt {b c} a^{3} c^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (15 \, b^{3} c^{2} x^{5} + 25 \, a b^{2} c^{2} x^{3} + 4 \, a^{2} b c^{2} x\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{195 \, b^{2}} \]
2/195*(12*sqrt(b*c)*a^3*c^2*weierstrassZeta(-4*a/b, 0, weierstrassPInverse (-4*a/b, 0, x)) + (15*b^3*c^2*x^5 + 25*a*b^2*c^2*x^3 + 4*a^2*b*c^2*x)*sqrt (b*x^2 + a)*sqrt(c*x))/b^2
Result contains complex when optimal does not.
Time = 8.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.14 \[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \]
a**(3/2)*c**(5/2)*x**(7/2)*gamma(7/4)*hyper((-3/2, 7/4), (11/4,), b*x**2*e xp_polar(I*pi)/a)/(2*gamma(11/4))
\[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {5}{2}} \,d x } \]
\[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx=\int {\left (c\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \]