Integrand size = 20, antiderivative size = 91 \[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}-\frac {\sqrt {a} \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\sqrt {b} \left (a-b x^2\right )^{3/4}} \] Output:
-c*(c*x)^(1/2)*(-b*x^2+a)^(1/4)/b-a^(1/2)*(1-a/b/x^2)^(3/4)*(c*x)^(3/2)*In verseJacobiAM(1/2*arccsc(b^(1/2)*x/a^(1/2)),2^(1/2))/b^(1/2)/(-b*x^2+a)^(3 /4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\frac {c \sqrt {c x} \left (-a+b x^2+a \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {b x^2}{a}\right )\right )}{b \left (a-b x^2\right )^{3/4}} \] Input:
Integrate[(c*x)^(3/2)/(a - b*x^2)^(3/4),x]
Output:
(c*Sqrt[c*x]*(-a + b*x^2 + a*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^2)/a]))/(b*(a - b*x^2)^(3/4))
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {262, 266, 768, 858, 807, 230}
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.
\(\displaystyle \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {a c^2 \int \frac {1}{\sqrt {c x} \left (a-b x^2\right )^{3/4}}dx}{2 b}-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {a c \int \frac {1}{\left (a-b x^2\right )^{3/4}}d\sqrt {c x}}{b}-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {a c (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}}d\sqrt {c x}}{b \left (a-b x^2\right )^{3/4}}-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {a c (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} \int \frac {1}{\sqrt {c x} \left (1-\frac {a c^4 x^2}{b}\right )^{3/4}}d\frac {1}{\sqrt {c x}}}{b \left (a-b x^2\right )^{3/4}}-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {a c (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} \int \frac {1}{\left (1-\frac {a c^3 x}{b}\right )^{3/4}}d(c x)}{2 b \left (a-b x^2\right )^{3/4}}-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}\) |
\(\Big \downarrow \) 230 |
\(\displaystyle -\frac {\sqrt {a} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a} c^2 x}{\sqrt {b}}\right ),2\right )}{\sqrt {b} \left (a-b x^2\right )^{3/4}}-\frac {c \sqrt {c x} \sqrt [4]{a-b x^2}}{b}\) |
Input:
Int[(c*x)^(3/2)/(a - b*x^2)^(3/4),x]
Output:
-((c*Sqrt[c*x]*(a - b*x^2)^(1/4))/b) - (Sqrt[a]*(1 - a/(b*x^2))^(3/4)*(c*x )^(3/2)*EllipticF[ArcSin[(Sqrt[a]*c^2*x)/Sqrt[b]]/2, 2])/(Sqrt[b]*(a - b*x ^2)^(3/4))
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] ))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
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[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int \frac {\left (c x \right )^{\frac {3}{2}}}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]
Input:
int((c*x)^(3/2)/(-b*x^2+a)^(3/4),x)
Output:
int((c*x)^(3/2)/(-b*x^2+a)^(3/4),x)
\[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate((c*x)^(3/2)/(-b*x^2+a)^(3/4),x, algorithm="fricas")
Output:
integral(-(-b*x^2 + a)^(1/4)*sqrt(c*x)*c*x/(b*x^2 - a), x)
Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.51 \[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\frac {c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((c*x)**(3/2)/(-b*x**2+a)**(3/4),x)
Output:
c**(3/2)*x**(5/2)*gamma(5/4)*hyper((3/4, 5/4), (9/4,), b*x**2*exp_polar(2* I*pi)/a)/(2*a**(3/4)*gamma(9/4))
\[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate((c*x)^(3/2)/(-b*x^2+a)^(3/4),x, algorithm="maxima")
Output:
integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4), x)
\[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate((c*x)^(3/2)/(-b*x^2+a)^(3/4),x, algorithm="giac")
Output:
integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4), x)
Timed out. \[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\int \frac {{\left (c\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^{3/4}} \,d x \] Input:
int((c*x)^(3/2)/(a - b*x^2)^(3/4),x)
Output:
int((c*x)^(3/2)/(a - b*x^2)^(3/4), x)
\[ \int \frac {(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx=\frac {\sqrt {c}\, c \left (-2 \sqrt {x}\, \left (-b \,x^{2}+a \right )^{\frac {1}{4}}+\left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{\sqrt {x}\, a -\sqrt {x}\, b \,x^{2}}d x \right ) a \right )}{2 b} \] Input:
int((c*x)^(3/2)/(-b*x^2+a)^(3/4),x)
Output:
(sqrt(c)*c*( - 2*sqrt(x)*(a - b*x**2)**(1/4) + int((a - b*x**2)**(1/4)/(sq rt(x)*a - sqrt(x)*b*x**2),x)*a))/(2*b)