\(\int \frac {(e x)^{7/2} (c+d x^2)}{(a+b x^2)^{9/4}} \, dx\) [1126]

Optimal result

Integrand size = 26, antiderivative size = 221 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {(4 b c-9 a d) e^{7/2} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {(4 b c-9 a d) e^{7/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}} \]

[Out]

2/5*(-a*d+b*c)*(e*x)^(9/2)/a/b/e/(b*x^2+a)^(5/4)-1/10*(-9*a*d+4*b*c)*e*(e*x)^(5/2)/a/b^2/(b*x^2+a)^(1/4)+1/4*(
-9*a*d+4*b*c)*e^(7/2)*arctan(b^(1/4)*(e*x)^(1/2)/(b*x^2+a)^(1/4)/e^(1/2))/b^(13/4)+1/4*(-9*a*d+4*b*c)*e^(7/2)*
arctanh(b^(1/4)*(e*x)^(1/2)/(b*x^2+a)^(1/4)/e^(1/2))/b^(13/4)-1/2*(-9*a*d+4*b*c)*e^3*(e*x)^(1/2)/b^3/(b*x^2+a)
^(1/4)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {468, 291, 294, 335, 246, 218, 214, 211} \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {e^{7/2} (4 b c-9 a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {e^{7/2} (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac {e^3 \sqrt {e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

[In]

Int[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(2*(b*c - a*d)*(e*x)^(9/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((4*b*c - 9*a*d)*e^3*Sqrt[e*x])/(2*b^3*(a + b*x^2)^(
1/4)) - ((4*b*c - 9*a*d)*e*(e*x)^(5/2))/(10*a*b^2*(a + b*x^2)^(1/4)) + ((4*b*c - 9*a*d)*e^(7/2)*ArcTan[(b^(1/4
)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(13/4)) + ((4*b*c - 9*a*d)*e^(7/2)*ArcTanh[(b^(1/4)*Sqrt[e*x])
/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(13/4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 291

Int[((c_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2)^(5/4), x_Symbol] :> Simp[2*c*((c*x)^(m - 1)/(b*(2*m - 3)*(a + b*x^
2)^(1/4))), x] - Dist[2*a*c^2*((m - 1)/(b*(2*m - 3))), Int[(c*x)^(m - 2)/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a
, b, c}, x] && PosQ[b/a] && IntegerQ[2*m] && GtQ[m, 3/2]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& 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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rubi steps \begin{align*} \text {integral}& = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {\left (2 \left (-2 b c+\frac {9 a d}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a b} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^2\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{4 b^2} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{4 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{2 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^4\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^3}+\frac {\left ((4 b c-9 a d) e^4\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {(4 b c-9 a d) e^{7/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {(4 b c-9 a d) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.90 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {(e x)^{7/2} \left (\frac {2 \sqrt [4]{b} \sqrt {x} \left (45 a^2 d+b^2 x^2 \left (-24 c+5 d x^2\right )+a b \left (-20 c+54 d x^2\right )\right )}{\left (a+b x^2\right )^{5/4}}+5 (4 b c-9 a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+5 (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{20 b^{13/4} x^{7/2}} \]

[In]

Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

((e*x)^(7/2)*((2*b^(1/4)*Sqrt[x]*(45*a^2*d + b^2*x^2*(-24*c + 5*d*x^2) + a*b*(-20*c + 54*d*x^2)))/(a + b*x^2)^
(5/4) + 5*(4*b*c - 9*a*d)*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + 5*(4*b*c - 9*a*d)*ArcTanh[(b^(1/4)*Sqr
t[x])/(a + b*x^2)^(1/4)]))/(20*b^(13/4)*x^(7/2))

Maple [F]

\[\int \frac {\left (e x \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}d x\]

[In]

int((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

[Out]

int((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 904, normalized size of antiderivative = 4.09 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {4 \, {\left (5 \, b^{2} d e^{3} x^{4} - 6 \, {\left (4 \, b^{2} c - 9 \, a b d\right )} e^{3} x^{2} - 5 \, {\left (4 \, a b c - 9 \, a^{2} d\right )} e^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} + 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} + {\left (b^{4} x^{2} + a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} - {\left (b^{4} x^{2} + a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 5 \, {\left (-i \, b^{5} x^{4} - 2 i \, a b^{4} x^{2} - i \, a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} + {\left (i \, b^{4} x^{2} + i \, a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 5 \, {\left (i \, b^{5} x^{4} + 2 i \, a b^{4} x^{2} + i \, a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} + {\left (-i \, b^{4} x^{2} - i \, a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right )}{40 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]

[In]

integrate((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="fricas")

[Out]

1/40*(4*(5*b^2*d*e^3*x^4 - 6*(4*b^2*c - 9*a*b*d)*e^3*x^2 - 5*(4*a*b*c - 9*a^2*d)*e^3)*(b*x^2 + a)^(3/4)*sqrt(e
*x) + 5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*
b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4)*log(-((b*x^2 + a)^(3/4)*(4*b*c - 9*a*d)*sqrt(e*x)*e^3 + (b^4*x^2 + a*
b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1
/4))/(b*x^2 + a)) - 5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^
2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4)*log(-((b*x^2 + a)^(3/4)*(4*b*c - 9*a*d)*sqrt(e*x)*e^3 -
(b^4*x^2 + a*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)
*e^14/b^13)^(1/4))/(b*x^2 + a)) - 5*(-I*b^5*x^4 - 2*I*a*b^4*x^2 - I*a^2*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d
+ 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4)*log(-((b*x^2 + a)^(3/4)*(4*b*c - 9
*a*d)*sqrt(e*x)*e^3 + (I*b^4*x^2 + I*a*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^
3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4))/(b*x^2 + a)) - 5*(I*b^5*x^4 + 2*I*a*b^4*x^2 + I*a^2*b^3)*((256*b^4
*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4)*log(-((b*x
^2 + a)^(3/4)*(4*b*c - 9*a*d)*sqrt(e*x)*e^3 + (-I*b^4*x^2 - I*a*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a
^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4))/(b*x^2 + a)))/(b^5*x^4 + 2*a*b^4*x^2 + a^
2*b^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4), x)

Giac [F]

\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \]

[In]

int(((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x)

[Out]

int(((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4), x)