Integrand size = 26, antiderivative size = 192 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {5 (2 b c-3 a d) e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac {(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}+\frac {5 \sqrt {a} (2 b c-3 a d) e^2 \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{6 b^{5/2} \left (a+b x^2\right )^{3/4}} \]
2/3*(-a*d+b*c)*(e*x)^(9/2)/a/b/e/(b*x^2+a)^(3/4)-1/3*(-3*a*d+2*b*c)*e*(e*x )^(5/2)*(b*x^2+a)^(1/4)/a/b^2+5/6*(-3*a*d+2*b*c)*e^2*(1+a/b/x^2)^(3/4)*(e* x)^(3/2)*(cos(1/2*arccot(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccot(x*b^( 1/2)/a^(1/2)))*EllipticF(sin(1/2*arccot(x*b^(1/2)/a^(1/2))),2^(1/2))*a^(1/ 2)/b^(5/2)/(b*x^2+a)^(3/4)+5/6*(-3*a*d+2*b*c)*e^3*(b*x^2+a)^(1/4)*(e*x)^(1 /2)/b^3
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.57 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {e^3 \sqrt {e x} \left (-15 a^2 d+a b \left (10 c-9 d x^2\right )+2 b^2 x^2 \left (3 c+d x^2\right )+5 a (-2 b c+3 a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{6 b^3 \left (a+b x^2\right )^{3/4}} \]
(e^3*Sqrt[e*x]*(-15*a^2*d + a*b*(10*c - 9*d*x^2) + 2*b^2*x^2*(3*c + d*x^2) + 5*a*(-2*b*c + 3*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(6*b^3*(a + b*x^2)^(3/4))
Time = 0.36 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {362, 262, 262, 266, 768, 858, 807, 229}
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.
\(\displaystyle \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx\) |
\(\Big \downarrow \) 362 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \int \frac {(e x)^{7/2}}{\left (b x^2+a\right )^{3/4}}dx}{a b}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \int \frac {(e x)^{3/2}}{\left (b x^2+a\right )^{3/4}}dx}{6 b}\right )}{a b}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \left (\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}-\frac {a e^2 \int \frac {1}{\sqrt {e x} \left (b x^2+a\right )^{3/4}}dx}{2 b}\right )}{6 b}\right )}{a b}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \left (\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}-\frac {a e \int \frac {1}{\left (b x^2+a\right )^{3/4}}d\sqrt {e x}}{b}\right )}{6 b}\right )}{a b}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \left (\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}-\frac {a e (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4} (e x)^{3/2}}d\sqrt {e x}}{b \left (a+b x^2\right )^{3/4}}\right )}{6 b}\right )}{a b}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \left (\frac {a e (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \int \frac {1}{\sqrt {e x} \left (\frac {a x^2 e^4}{b}+1\right )^{3/4}}d\frac {1}{\sqrt {e x}}}{b \left (a+b x^2\right )^{3/4}}+\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}\right )}{6 b}\right )}{a b}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \left (\frac {a e (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \int \frac {1}{\left (\frac {a x e^3}{b}+1\right )^{3/4}}d(e x)}{2 b \left (a+b x^2\right )^{3/4}}+\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}\right )}{6 b}\right )}{a b}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-3 a d) \left (\frac {e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b}-\frac {5 a e^2 \left (\frac {\sqrt {a} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} e^2 x}{\sqrt {b}}\right ),2\right )}{\sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}\right )}{6 b}\right )}{a b}\) |
(2*(b*c - a*d)*(e*x)^(9/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - ((2*b*c - 3*a*d) *((e*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(3*b) - (5*a*e^2*((e*Sqrt[e*x]*(a + b* x^2)^(1/4))/b + (Sqrt[a]*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcTa n[(Sqrt[a]*e^2*x)/Sqrt[b]]/2, 2])/(Sqrt[b]*(a + b*x^2)^(3/4))))/(6*b)))/(a *b)
3.12.21.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
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]
\[\int \frac {\left (e x \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {7}{4}}}d x\]
\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}} \,d x } \]
integral((d*e^3*x^5 + c*e^3*x^3)*(b*x^2 + a)^(1/4)*sqrt(e*x)/(b^2*x^4 + 2* a*b*x^2 + a^2), x)
Result contains complex when optimal does not.
Time = 149.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.49 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {c e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {13}{4}\right )} + \frac {d e^{\frac {7}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {17}{4}\right )} \]
c*e**(7/2)*x**(9/2)*gamma(9/4)*hyper((7/4, 9/4), (13/4,), b*x**2*exp_polar (I*pi)/a)/(2*a**(7/4)*gamma(13/4)) + d*e**(7/2)*x**(13/2)*gamma(13/4)*hype r((7/4, 13/4), (17/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(7/4)*gamma(17/4))
\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}} \,d x } \]
\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{7/4}} \,d x \]