\(\int \frac {(e x)^{3/2} (c+d x^2)}{(a+b x^2)^{7/4}} \, dx\) [467]

Optimal result

Integrand size = 26, antiderivative size = 133 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 (b c-a d) e \sqrt {e x}}{3 b^2 \left (a+b x^2\right )^{3/4}}+\frac {d e \sqrt {e x} \sqrt [4]{a+b x^2}}{b^2}-\frac {(2 b c-5 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {a} b^{3/2} \left (a+b x^2\right )^{3/4}} \] Output:

-2/3*(-a*d+b*c)*e*(e*x)^(1/2)/b^2/(b*x^2+a)^(3/4)+d*e*(e*x)^(1/2)*(b*x^2+a 
)^(1/4)/b^2-1/3*(-5*a*d+2*b*c)*(1+a/b/x^2)^(3/4)*(e*x)^(3/2)*InverseJacobi 
AM(1/2*arccot(b^(1/2)*x/a^(1/2)),2^(1/2))/a^(1/2)/b^(3/2)/(b*x^2+a)^(3/4)
 

Mathematica [C] (verified)

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

Time = 10.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {e \sqrt {e x} \left (-2 b c+5 a d+3 b d x^2+(2 b c-5 a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 b^2 \left (a+b x^2\right )^{3/4}} \] Input:

Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {362, 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.

\(\displaystyle \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \int \frac {(e x)^{3/2}}{\left (b x^2+a\right )^{3/4}}dx}{3 a b}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \left (\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}-\frac {a e^2 \int \frac {1}{\sqrt {e x} \left (b x^2+a\right )^{3/4}}dx}{2 b}\right )}{3 a b}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \left (\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}-\frac {a e \int \frac {1}{\left (b x^2+a\right )^{3/4}}d\sqrt {e x}}{b}\right )}{3 a b}\)

\(\Big \downarrow \) 768

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \left (\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}-\frac {a e (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4} (e x)^{3/2}}d\sqrt {e x}}{b \left (a+b x^2\right )^{3/4}}\right )}{3 a b}\)

\(\Big \downarrow \) 858

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \left (\frac {a e (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \int \frac {1}{\sqrt {e x} \left (\frac {a x^2 e^4}{b}+1\right )^{3/4}}d\frac {1}{\sqrt {e x}}}{b \left (a+b x^2\right )^{3/4}}+\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}\right )}{3 a b}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \left (\frac {a e (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \int \frac {1}{\left (\frac {a x e^3}{b}+1\right )^{3/4}}d(e x)}{2 b \left (a+b x^2\right )^{3/4}}+\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}\right )}{3 a b}\)

\(\Big \downarrow \) 229

\(\displaystyle \frac {2 (e x)^{5/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(2 b c-5 a d) \left (\frac {\sqrt {a} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} e^2 x}{\sqrt {b}}\right ),2\right )}{\sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {e \sqrt {e x} \sqrt [4]{a+b x^2}}{b}\right )}{3 a b}\)

Input:

Int[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]
 

Output:

(2*(b*c - a*d)*(e*x)^(5/2))/(3*a*b*e*(a + b*x^2)^(3/4)) - ((2*b*c - 5*a*d) 
*((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[ArcTan[(Sqrt[a]*e^2*x)/Sqrt[b]]/2, 2])/(Sqrt[b]*(a + b*x^ 
2)^(3/4))))/(3*a*b)
 

Defintions of rubi rules used

rule 229

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 262

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]
 

rule 266

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]
 

rule 362

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)]))
 

rule 768

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]
 

rule 807

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]
 

rule 858

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]
 

Maple [F]

\[\int \frac {\left (e x \right )^{\frac {3}{2}} \left (x^{2} d +c \right )}{\left (b \,x^{2}+a \right )^{\frac {7}{4}}}d x\]

Input:

int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
 

Output:

int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
 

Fricas [F]

\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x, algorithm="fricas")
 

Output:

integral((d*e*x^3 + c*e*x)*(b*x^2 + a)^(1/4)*sqrt(e*x)/(b^2*x^4 + 2*a*b*x^ 
2 + a^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 19.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {c e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {13}{4}\right )} \] Input:

integrate((e*x)**(3/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
 

Output:

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

Maxima [F]

\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(7/4), x)
 

Giac [F]

\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(7/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{7/4}} \,d x \] Input:

int(((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x)
 

Output:

int(((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(7/4), x)
 

Reduce [F]

\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\sqrt {e}\, e \left (\left (\int \frac {\sqrt {x}\, x^{3}}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} a +\left (b \,x^{2}+a \right )^{\frac {3}{4}} b \,x^{2}}d x \right ) d +\left (\int \frac {\sqrt {x}\, x}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} a +\left (b \,x^{2}+a \right )^{\frac {3}{4}} b \,x^{2}}d x \right ) c \right ) \] Input:

int((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
 

Output:

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