Integrand size = 19, antiderivative size = 215 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=-\frac {2 a \sqrt {a+b x^2}}{15 c (c x)^{15/2}}-\frac {34 b \sqrt {a+b x^2}}{165 c^3 (c x)^{11/2}}-\frac {8 b^2 \sqrt {a+b x^2}}{385 a c^5 (c x)^{7/2}}+\frac {8 b^3 \sqrt {a+b x^2}}{231 a^2 c^7 (c x)^{3/2}}+\frac {4 b^{15/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{231 a^{9/4} c^{17/2} \sqrt {a+b x^2}} \] Output:
-2/15*a*(b*x^2+a)^(1/2)/c/(c*x)^(15/2)-34/165*b*(b*x^2+a)^(1/2)/c^3/(c*x)^ (11/2)-8/385*b^2*(b*x^2+a)^(1/2)/a/c^5/(c*x)^(7/2)+8/231*b^3*(b*x^2+a)^(1/ 2)/a^2/c^7/(c*x)^(3/2)+4/231*b^(15/4)*(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))/a^(9/4)/c^(17/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.27 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=-\frac {2 a x \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {15}{4},-\frac {3}{2},-\frac {11}{4},-\frac {b x^2}{a}\right )}{15 (c x)^{17/2} \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[(a + b*x^2)^(3/2)/(c*x)^(17/2),x]
Output:
(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-15/4, -3/2, -11/4, -((b*x^2)/a) ])/(15*(c*x)^(17/2)*Sqrt[1 + (b*x^2)/a])
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {247, 247, 264, 264, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {2 b \int \frac {\sqrt {b x^2+a}}{(c x)^{13/2}}dx}{5 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{15 c (c x)^{15/2}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {2 b \left (\frac {2 b \int \frac {1}{(c x)^{9/2} \sqrt {b x^2+a}}dx}{11 c^2}-\frac {2 \sqrt {a+b x^2}}{11 c (c x)^{11/2}}\right )}{5 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{15 c (c x)^{15/2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {2 b \left (\frac {2 b \left (-\frac {5 b \int \frac {1}{(c x)^{5/2} \sqrt {b x^2+a}}dx}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\right )}{11 c^2}-\frac {2 \sqrt {a+b x^2}}{11 c (c x)^{11/2}}\right )}{5 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{15 c (c x)^{15/2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {2 b \left (\frac {2 b \left (-\frac {5 b \left (-\frac {b \int \frac {1}{\sqrt {c x} \sqrt {b x^2+a}}dx}{3 a c^2}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\right )}{11 c^2}-\frac {2 \sqrt {a+b x^2}}{11 c (c x)^{11/2}}\right )}{5 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{15 c (c x)^{15/2}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 b \left (\frac {2 b \left (-\frac {5 b \left (-\frac {2 b \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{3 a c^3}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\right )}{11 c^2}-\frac {2 \sqrt {a+b x^2}}{11 c (c x)^{11/2}}\right )}{5 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{15 c (c x)^{15/2}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 b \left (\frac {2 b \left (-\frac {5 b \left (-\frac {b^{3/4} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{3 a^{5/4} c^{7/2} \sqrt {a+b x^2}}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\right )}{11 c^2}-\frac {2 \sqrt {a+b x^2}}{11 c (c x)^{11/2}}\right )}{5 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{15 c (c x)^{15/2}}\) |
Input:
Int[(a + b*x^2)^(3/2)/(c*x)^(17/2),x]
Output:
(-2*(a + b*x^2)^(3/2))/(15*c*(c*x)^(15/2)) + (2*b*((-2*Sqrt[a + b*x^2])/(1 1*c*(c*x)^(11/2)) + (2*b*((-2*Sqrt[a + b*x^2])/(7*a*c*(c*x)^(7/2)) - (5*b* ((-2*Sqrt[a + b*x^2])/(3*a*c*(c*x)^(3/2)) - (b^(3/4)*(Sqrt[a]*c + Sqrt[b]* c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x)^2]*EllipticF[2*Arc Tan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(3*a^(5/4)*c^(7/2)*Sqrt[ a + b*x^2])))/(7*a*c^2)))/(11*c^2)))/(5*c^2)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && 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 = 2.75 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\frac {4 \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b^{3} x^{7}}{231}+\frac {8 b^{4} x^{8}}{231}+\frac {16 a \,b^{3} x^{6}}{1155}-\frac {262 a^{2} b^{2} x^{4}}{1155}-\frac {56 a^{3} b \,x^{2}}{165}-\frac {2 a^{4}}{15}}{\sqrt {b \,x^{2}+a}\, x^{7} a^{2} c^{8} \sqrt {c x}}\) | \(162\) |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-20 b^{3} x^{6}+12 a \,b^{2} x^{4}+119 a^{2} b \,x^{2}+77 a^{3}\right )}{1155 x^{7} a^{2} c^{8} \sqrt {c x}}+\frac {4 b^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{231 a^{2} \sqrt {b c \,x^{3}+a c x}\, c^{8} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(201\) |
elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 a \sqrt {b c \,x^{3}+a c x}}{15 c^{9} x^{8}}-\frac {34 b \sqrt {b c \,x^{3}+a c x}}{165 c^{9} x^{6}}-\frac {8 b^{2} \sqrt {b c \,x^{3}+a c x}}{385 a \,c^{9} x^{4}}+\frac {8 b^{3} \sqrt {b c \,x^{3}+a c x}}{231 a^{2} c^{9} x^{2}}+\frac {4 b^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 a^{2} c^{8} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(243\) |
Input:
int((b*x^2+a)^(3/2)/(c*x)^(17/2),x,method=_RETURNVERBOSE)
Output:
2/1155/(b*x^2+a)^(1/2)/x^7*(10*(-a*b)^(1/2)*((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))* b^3*x^7+20*b^4*x^8+8*a*b^3*x^6-131*a^2*b^2*x^4-196*a^3*b*x^2-77*a^4)/a^2/c ^8/(c*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=\frac {2 \, {\left (20 \, \sqrt {b c} b^{3} x^{8} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (20 \, b^{3} x^{6} - 12 \, a b^{2} x^{4} - 119 \, a^{2} b x^{2} - 77 \, a^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{1155 \, a^{2} c^{9} x^{8}} \] Input:
integrate((b*x^2+a)^(3/2)/(c*x)^(17/2),x, algorithm="fricas")
Output:
2/1155*(20*sqrt(b*c)*b^3*x^8*weierstrassPInverse(-4*a/b, 0, x) + (20*b^3*x ^6 - 12*a*b^2*x^4 - 119*a^2*b*x^2 - 77*a^3)*sqrt(b*x^2 + a)*sqrt(c*x))/(a^ 2*c^9*x^8)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/2)/(c*x)**(17/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {17}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(c*x)^(17/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)/(c*x)^(17/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {17}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(c*x)^(17/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)/(c*x)^(17/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c\,x\right )}^{17/2}} \,d x \] Input:
int((a + b*x^2)^(3/2)/(c*x)^(17/2),x)
Output:
int((a + b*x^2)^(3/2)/(c*x)^(17/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{17/2}} \, dx=\frac {2 \sqrt {c}\, \left (-7 \sqrt {b \,x^{2}+a}\, a -13 \sqrt {b \,x^{2}+a}\, b \,x^{2}+6 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{11}+a \,x^{9}}d x \right ) a^{2} x^{7}\right )}{117 \sqrt {x}\, c^{9} x^{7}} \] Input:
int((b*x^2+a)^(3/2)/(c*x)^(17/2),x)
Output:
(2*sqrt(c)*( - 7*sqrt(a + b*x**2)*a - 13*sqrt(a + b*x**2)*b*x**2 + 6*sqrt( x)*int((sqrt(x)*sqrt(a + b*x**2))/(a*x**9 + b*x**11),x)*a**2*x**7))/(117*s qrt(x)*c**9*x**7)