\(\int \frac {c+d x^2}{(e x)^{9/2} (a+b x^2)^{3/4}} \, dx\) [1103]

Optimal result

Integrand size = 26, antiderivative size = 144 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {4 b^{3/2} (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}} \]

[Out]

-2/7*c*(b*x^2+a)^(1/4)/a/e/(e*x)^(7/2)+2/21*(-7*a*d+6*b*c)*(b*x^2+a)^(1/4)/a^2/e^3/(e*x)^(3/2)-4/21*b^(3/2)*(-
7*a*d+6*b*c)*(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^(5/2)/e^6/(b*x^2+a)^(3/4)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {464, 331, 335, 243, 342, 281, 237} \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {4 b^{3/2} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (6 b c-7 a d) \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac {2 \sqrt [4]{a+b x^2} (6 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}} \]

[In]

Int[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(7*a*e*(e*x)^(7/2)) + (2*(6*b*c - 7*a*d)*(a + b*x^2)^(1/4))/(21*a^2*e^3*(e*x)^(3/2))
- (4*b^(3/2)*(6*b*c - 7*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2
1*a^(5/2)*e^6*(a + b*x^2)^(3/4))

Rule 237

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]

Rule 243

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[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]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[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]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

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] && IntegerQ[m]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}-\frac {(6 b c-7 a d) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{7 a e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}+\frac {(2 b (6 b c-7 a d)) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{21 a^2 e^4} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}+\frac {(4 b (6 b c-7 a d)) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{21 a^2 e^5} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}+\frac {\left (4 b (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {e x}\right )}{21 a^2 e^5 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {\left (4 b (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {e x}}\right )}{21 a^2 e^5 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {\left (2 b (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{e x}\right )}{21 a^2 e^5 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {4 b^{3/2} (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.61 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 \sqrt {e x} \left (3 c \left (a+b x^2\right )+(-6 b c+7 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{4},\frac {1}{4},-\frac {b x^2}{a}\right )\right )}{21 a e^5 x^4 \left (a+b x^2\right )^{3/4}} \]

[In]

Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*Sqrt[e*x]*(3*c*(a + b*x^2) + (-6*b*c + 7*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/4, 3/4, 1/4,
-((b*x^2)/a)]))/(21*a*e^5*x^4*(a + b*x^2)^(3/4))

Maple [F]

\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]

[In]

int((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(3/4),x)

[Out]

int((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(3/4),x)

Fricas [F]

\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/4)*(d*x^2 + c)*sqrt(e*x)/(b*e^5*x^7 + a*e^5*x^5), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 100.43 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=- \frac {c {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{5 b^{\frac {3}{4}} e^{\frac {9}{2}} x^{5}} + \frac {d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]

[In]

integrate((d*x**2+c)/(e*x)**(9/2)/(b*x**2+a)**(3/4),x)

[Out]

-c*hyper((3/4, 5/2), (7/2,), a*exp_polar(I*pi)/(b*x**2))/(5*b**(3/4)*e**(9/2)*x**5) + d*gamma(-3/4)*hyper((-3/
4, 3/4), (1/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/4)*e**(9/2)*x**(3/2)*gamma(1/4))

Maxima [F]

\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(9/2)), x)

Giac [F]

\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(9/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{9/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]

[In]

int((c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)),x)

[Out]

int((c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(3/4)), x)