Integrand size = 19, antiderivative size = 184 \[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=-\frac {20 a^2 c^3 \sqrt {c x} \sqrt {a+b x^2}}{231 b^2}+\frac {4 a c (c x)^{5/2} \sqrt {a+b x^2}}{77 b}+\frac {2 (c x)^{9/2} \sqrt {a+b x^2}}{11 c}+\frac {10 a^{11/4} c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}} \] Output:
-20/231*a^2*c^3*(c*x)^(1/2)*(b*x^2+a)^(1/2)/b^2+4/77*a*c*(c*x)^(5/2)*(b*x^ 2+a)^(1/2)/b+2/11*(c*x)^(9/2)*(b*x^2+a)^(1/2)/c+10/231*a^(11/4)*c^(7/2)*(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^(9/4)/(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 = 103, normalized size of antiderivative = 0.56 \[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\frac {2 c^3 \sqrt {c x} \sqrt {a+b x^2} \left (\sqrt {1+\frac {b x^2}{a}} \left (-5 a^2+2 a b x^2+7 b^2 x^4\right )+5 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{77 b^2 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[(c*x)^(7/2)*Sqrt[a + b*x^2],x]
Output:
(2*c^3*Sqrt[c*x]*Sqrt[a + b*x^2]*(Sqrt[1 + (b*x^2)/a]*(-5*a^2 + 2*a*b*x^2 + 7*b^2*x^4) + 5*a^2*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/(77 *b^2*Sqrt[1 + (b*x^2)/a])
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {248, 262, 262, 266, 761}
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.
| \(\displaystyle \int (c x)^{7/2} \sqrt {a+b x^2} \, dx\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {2}{11} a \int \frac {(c x)^{7/2}}{\sqrt {b x^2+a}}dx+\frac {2 (c x)^{9/2} \sqrt {a+b x^2}}{11 c}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2}{11} a \left (\frac {2 c (c x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {5 a c^2 \int \frac {(c x)^{3/2}}{\sqrt {b x^2+a}}dx}{7 b}\right )+\frac {2 (c x)^{9/2} \sqrt {a+b x^2}}{11 c}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2}{11} a \left (\frac {2 c (c x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {5 a c^2 \left (\frac {2 c \sqrt {c x} \sqrt {a+b x^2}}{3 b}-\frac {a c^2 \int \frac {1}{\sqrt {c x} \sqrt {b x^2+a}}dx}{3 b}\right )}{7 b}\right )+\frac {2 (c x)^{9/2} \sqrt {a+b x^2}}{11 c}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2}{11} a \left (\frac {2 c (c x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {5 a c^2 \left (\frac {2 c \sqrt {c x} \sqrt {a+b x^2}}{3 b}-\frac {2 a c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{3 b}\right )}{7 b}\right )+\frac {2 (c x)^{9/2} \sqrt {a+b x^2}}{11 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2}{11} a \left (\frac {2 c (c x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {5 a c^2 \left (\frac {2 c \sqrt {c x} \sqrt {a+b x^2}}{3 b}-\frac {a^{3/4} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{3 b^{5/4} \sqrt {a+b x^2}}\right )}{7 b}\right )+\frac {2 (c x)^{9/2} \sqrt {a+b x^2}}{11 c}\) |
Input:
Int[(c*x)^(7/2)*Sqrt[a + b*x^2],x]
Output:
(2*(c*x)^(9/2)*Sqrt[a + b*x^2])/(11*c) + (2*a*((2*c*(c*x)^(5/2)*Sqrt[a + b *x^2])/(7*b) - (5*a*c^2*((2*c*Sqrt[c*x]*Sqrt[a + b*x^2])/(3*b) - (a^(3/4)* Sqrt[c]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sq rt[b]*c*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1 /2])/(3*b^(5/4)*Sqrt[a + b*x^2])))/(7*b)))/11
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]
Time = 0.64 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(\frac {2 c^{3} \sqrt {c x}\, \left (21 b^{4} x^{7}+5 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{3}+27 a \,b^{3} x^{5}-4 a^{2} b^{2} x^{3}-10 a^{3} b x \right )}{231 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(152\) |
| risch | \(-\frac {2 \left (-21 b^{2} x^{4}-6 a b \,x^{2}+10 a^{2}\right ) x \sqrt {b \,x^{2}+a}\, c^{4}}{231 b^{2} \sqrt {c x}}+\frac {10 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c^{4} \sqrt {c x \left (b \,x^{2}+a \right )}}{231 b^{3} \sqrt {b c \,x^{3}+a c x}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(188\) |
| elliptic | \(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {2 c^{3} x^{4} \sqrt {b c \,x^{3}+a c x}}{11}+\frac {4 a \,c^{3} x^{2} \sqrt {b c \,x^{3}+a c x}}{77 b}-\frac {20 a^{2} c^{3} \sqrt {b c \,x^{3}+a c x}}{231 b^{2}}+\frac {10 a^{3} c^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 b^{3} \sqrt {b c \,x^{3}+a c x}}\right )}{c x \sqrt {b \,x^{2}+a}}\) | \(221\) |
Input:
int((c*x)^(7/2)*(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/231*c^3/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)*(21*b^4*x^7+5*2^(1/2)*((b*x+(-a*b) ^(1/2))/(-a*b)^(1/2))^(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*b)^(1/2)*a^3+27*a*b^3*x^5-4*a^2*b^2*x^3-10*a^3*b*x)/b^3
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.41 \[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\frac {2 \, {\left (10 \, \sqrt {b c} a^{3} c^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (21 \, b^{3} c^{3} x^{4} + 6 \, a b^{2} c^{3} x^{2} - 10 \, a^{2} b c^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{231 \, b^{3}} \] Input:
integrate((c*x)^(7/2)*(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/231*(10*sqrt(b*c)*a^3*c^3*weierstrassPInverse(-4*a/b, 0, x) + (21*b^3*c^ 3*x^4 + 6*a*b^2*c^3*x^2 - 10*a^2*b*c^3)*sqrt(b*x^2 + a)*sqrt(c*x))/b^3
Result contains complex when optimal does not.
Time = 14.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.25 \[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\frac {\sqrt {a} c^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((c*x)**(7/2)*(b*x**2+a)**(1/2),x)
Output:
sqrt(a)*c**(7/2)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), b*x**2*ex p_polar(I*pi)/a)/(2*gamma(13/4))
\[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \left (c x\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((c*x)^(7/2)*(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(c*x)^(7/2), x)
\[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\int { \sqrt {b x^{2} + a} \left (c x\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((c*x)^(7/2)*(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(c*x)^(7/2), x)
Timed out. \[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\int {\left (c\,x\right )}^{7/2}\,\sqrt {b\,x^2+a} \,d x \] Input:
int((c*x)^(7/2)*(a + b*x^2)^(1/2),x)
Output:
int((c*x)^(7/2)*(a + b*x^2)^(1/2), x)
\[ \int (c x)^{7/2} \sqrt {a+b x^2} \, dx=\frac {2 \sqrt {c}\, c^{3} \left (-10 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2}+6 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a b \,x^{2}+21 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b^{2} x^{4}+5 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{3}+a x}d x \right ) a^{3}\right )}{231 b^{2}} \] Input:
int((c*x)^(7/2)*(b*x^2+a)^(1/2),x)
Output:
(2*sqrt(c)*c**3*( - 10*sqrt(x)*sqrt(a + b*x**2)*a**2 + 6*sqrt(x)*sqrt(a + b*x**2)*a*b*x**2 + 21*sqrt(x)*sqrt(a + b*x**2)*b**2*x**4 + 5*int((sqrt(x)* sqrt(a + b*x**2))/(a*x + b*x**3),x)*a**3))/(231*b**2)