Integrand size = 19, antiderivative size = 297 \[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}} \] Output:
4/15*a*(c*x)^(3/2)*(b*x^2+a)^(1/2)/c+8/15*a^2*(c*x)^(1/2)*(b*x^2+a)^(1/2)/ b^(1/2)/(a^(1/2)+b^(1/2)*x)+2/9*(c*x)^(3/2)*(b*x^2+a)^(3/2)/c-8/15*a^(9/4) *c^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*Ellip ticE(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))/b^(3/ 4)/(b*x^2+a)^(1/2)+4/15*a^(9/4)*c^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^ (1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*(c*x)^(1/2)/a^( 1/4)/c^(1/2)),1/2*2^(1/2))/b^(3/4)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.19 \[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\frac {2 a x \sqrt {c x} \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )}{3 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[Sqrt[c*x]*(a + b*x^2)^(3/2),x]
Output:
(2*a*x*Sqrt[c*x]*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^ 2)/a)])/(3*Sqrt[1 + (b*x^2)/a])
Time = 0.37 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {248, 248, 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.
\(\displaystyle \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {2}{3} a \int \sqrt {c x} \sqrt {b x^2+a}dx+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \int \frac {\sqrt {c x}}{\sqrt {b x^2+a}}dx+\frac {2 (c x)^{3/2} \sqrt {a+b x^2}}{5 c}\right )+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2}{3} a \left (\frac {4 a \int \frac {c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{5 c}+\frac {2 (c x)^{3/2} \sqrt {a+b x^2}}{5 c}\right )+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\sqrt {a} c \int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {a} c \sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{5 c}+\frac {2 (c x)^{3/2} \sqrt {a+b x^2}}{5 c}\right )+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{5 c}+\frac {2 (c x)^{3/2} \sqrt {a+b x^2}}{5 c}\right )+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{5 c}+\frac {2 (c x)^{3/2} \sqrt {a+b x^2}}{5 c}\right )+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {c^2 \sqrt {c x} \sqrt {a+b x^2}}{\sqrt {a} c+\sqrt {b} c x}}{\sqrt {b}}\right )}{5 c}+\frac {2 (c x)^{3/2} \sqrt {a+b x^2}}{5 c}\right )+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}\) |
Input:
Int[Sqrt[c*x]*(a + b*x^2)^(3/2),x]
Output:
(2*(c*x)^(3/2)*(a + b*x^2)^(3/2))/(9*c) + (2*a*((2*(c*x)^(3/2)*Sqrt[a + b* x^2])/(5*c) + (4*a*(-((-((c^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(Sqrt[a]*c + Sqrt [b]*c*x)) + (a^(1/4)*Sqrt[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)*S qrt[c]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqr t[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*c)))/3
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] :> 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 = 0.47 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {2 \sqrt {c x}\, \left (5 b^{3} x^{6}+12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+16 a \,b^{2} x^{4}+11 a^{2} b \,x^{2}\right )}{45 \sqrt {b \,x^{2}+a}\, b x}\) | \(218\) |
risch | \(\frac {2 x^{2} \left (5 b \,x^{2}+11 a \right ) \sqrt {b \,x^{2}+a}\, c}{45 \sqrt {c x}}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) c \sqrt {c x \left (b \,x^{2}+a \right )}}{15 b \sqrt {b c \,x^{3}+a c x}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(222\) |
elliptic | \(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {2 b \,x^{3} \sqrt {b c \,x^{3}+a c x}}{9}+\frac {22 a x \sqrt {b c \,x^{3}+a c x}}{45}+\frac {4 a^{2} c \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 b \sqrt {b c \,x^{3}+a c x}}\right )}{c x \sqrt {b \,x^{2}+a}}\) | \(235\) |
Input:
int((c*x)^(1/2)*(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/45*(c*x)^(1/2)/(b*x^2+a)^(1/2)/b*(5*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)*(-b/(-a*b) ^(1/2)*x)^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1 /2))*a^3-6*((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)*(-b/(-a*b)^(1/2)*x)^(1/2)*EllipticF(((b*x+(-a*b) ^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^3+16*a*b^2*x^4+11*a^2*b*x^2)/x
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.21 \[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (12 \, \sqrt {b c} a^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, b^{2} x^{3} + 11 \, a b x\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{45 \, b} \] Input:
integrate((c*x)^(1/2)*(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-2/45*(12*sqrt(b*c)*a^2*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4* a/b, 0, x)) - (5*b^2*x^3 + 11*a*b*x)*sqrt(b*x^2 + a)*sqrt(c*x))/b
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.15 \[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((c*x)**(1/2)*(b*x**2+a)**(3/2),x)
Output:
a**(3/2)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((-3/2, 3/4), (7/4,), b*x**2*exp _polar(I*pi)/a)/(2*gamma(7/4))
\[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x} \,d x } \] Input:
integrate((c*x)^(1/2)*(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(c*x), x)
\[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x} \,d x } \] Input:
integrate((c*x)^(1/2)*(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(c*x), x)
Timed out. \[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\int \sqrt {c\,x}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \] Input:
int((c*x)^(1/2)*(a + b*x^2)^(3/2),x)
Output:
int((c*x)^(1/2)*(a + b*x^2)^(3/2), x)
\[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c}\, \left (11 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a x +5 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b \,x^{3}+6 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{2}+a}d x \right ) a^{2}\right )}{45} \] Input:
int((c*x)^(1/2)*(b*x^2+a)^(3/2),x)
Output:
(2*sqrt(c)*(11*sqrt(x)*sqrt(a + b*x**2)*a*x + 5*sqrt(x)*sqrt(a + b*x**2)*b *x**3 + 6*int((sqrt(x)*sqrt(a + b*x**2))/(a + b*x**2),x)*a**2))/45