Integrand size = 26, antiderivative size = 349 \[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {3 d e^2 \sqrt {e x} \sqrt {c+d x}}{2 b^2}-\frac {e (e x)^{3/2} (c+d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac {3 \left (b c^2-3 \sqrt {-a} \sqrt {b} c d-2 a d^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 \sqrt [4]{-a} b^{5/2} \sqrt {\sqrt {b} c-\sqrt {-a} d}}+\frac {3 c \sqrt {d} e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b^2}-\frac {3 \left (b c^2+3 \sqrt {-a} \sqrt {b} c d-2 a d^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 \sqrt [4]{-a} b^{5/2} \sqrt {\sqrt {b} c+\sqrt {-a} d}} \] Output:
3/2*d*e^2*(e*x)^(1/2)*(d*x+c)^(1/2)/b^2-1/2*e*(e*x)^(3/2)*(d*x+c)^(3/2)/b/ (b*x^2+a)+3/4*(b*c^2-3*(-a)^(1/2)*b^(1/2)*c*d-2*a*d^2)*e^(5/2)*arctan((b^( 1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/( -a)^(1/4)/b^(5/2)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)+3*c*d^(1/2)*e^(5/2)*arcta nh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/b^2-3/4*(b*c^2+3*(-a)^(1/2)* b^(1/2)*c*d-2*a*d^2)*e^(5/2)*arctanh((b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x)^ (1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(1/4)/b^(5/2)/(b^(1/2)*c+(-a) ^(1/2)*d)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.74 (sec) , antiderivative size = 1073, normalized size of antiderivative = 3.07 \[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[((e*x)^(5/2)*(c + d*x)^(3/2))/(a + b*x^2)^2,x]
Output:
((e*x)^(5/2)*(8*a*Sqrt[x]*Sqrt[c + d*x]*(3*a*d + b*x*(-c + 2*d*x)) + 96*a* c*Sqrt[d]*(a + b*x^2)*ArcTanh[(Sqrt[d]*Sqrt[x])/(-Sqrt[c] + Sqrt[c + d*x]) ] + c*(a + b*x^2)*RootSum[a*d^4 - 4*a*d^3*#1^2 + 16*b*c^2*#1^4 + 6*a*d^2*# 1^4 - 4*a*d*#1^6 + a*#1^8 & , (256*b^2*c^4*Log[x] - 384*a*b*c^2*d^2*Log[x] - a^2*d^4*Log[x] - 512*b^2*c^4*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] + 768*a*b*c^2*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] + 2*a^2*d^4* Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] - 116*a*b*c^2*d*Log[x]*#1^2 - 5 *a^2*d^3*Log[x]*#1^2 + 232*a*b*c^2*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x ]*#1]*#1^2 + 10*a^2*d^3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 - 12*a*b*c^2*Log[x]*#1^4 + 5*a^2*d^2*Log[x]*#1^4 + 24*a*b*c^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - 10*a^2*d^2*Log[-Sqrt[c] + Sqrt[c + d*x ] - Sqrt[x]*#1]*#1^4 + a^2*d*Log[x]*#1^6 - 2*a^2*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6)/(a*d^3*#1 - 8*b*c^2*#1^3 - 3*a*d^2*#1^3 + 3*a*d* #1^5 - a*#1^7) & ] - 8*c*(a + b*x^2)*RootSum[a*d^4 - 4*a*d^3*#1^2 + 16*b*c ^2*#1^4 + 6*a*d^2*#1^4 - 4*a*d*#1^6 + a*#1^8 & , (32*b^2*c^4*Log[x] - 48*a *b*c^2*d^2*Log[x] + a^2*d^4*Log[x] - 64*b^2*c^4*Log[-Sqrt[c] + Sqrt[c + d* x] - Sqrt[x]*#1] + 96*a*b*c^2*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*# 1] - 2*a^2*d^4*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] - 16*a*b*c^2*d*L og[x]*#1^2 - a^2*d^3*Log[x]*#1^2 + 32*a*b*c^2*d*Log[-Sqrt[c] + Sqrt[c + d* x] - Sqrt[x]*#1]*#1^2 + 2*a^2*d^3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x...
Leaf count is larger than twice the leaf count of optimal. \(1480\) vs. \(2(349)=698\).
Time = 5.40 (sec) , antiderivative size = 1480, normalized size of antiderivative = 4.24, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {615, 2009}
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.
| \(\displaystyle \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (-\frac {b (e x)^{5/2} (c+d x)^{3/2}}{2 a \left (-a b-b^2 x^2\right )}-\frac {b (e x)^{5/2} (c+d x)^{3/2}}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b (e x)^{5/2} (c+d x)^{3/2}}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) c^3}{16 a b d^{3/2}}-\frac {7 e^2 \sqrt {e x} \sqrt {c+d x} c^2}{16 a b d}+\frac {7 e (e x)^{3/2} \sqrt {c+d x} c}{24 a b}-\frac {2 e (e x)^{3/2} (c+d x)^{3/2}}{3 a b}-\frac {\left (\sqrt {b} c-8 \sqrt {-a} d\right ) e^2 \sqrt {e x} (c+d x)^{3/2}}{16 a b^{3/2} d}-\frac {\left (\sqrt {b} c+8 \sqrt {-a} d\right ) e^2 \sqrt {e x} (c+d x)^{3/2}}{16 a b^{3/2} d}-\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 a \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\left (5 b c^2-13 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 \sqrt [4]{-a} b^{5/2} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 \sqrt [4]{-a} b^{5/2} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (\sqrt {b} c+2 \sqrt {-a} d\right ) \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{4 a b^{5/2} d^{3/2}}+\frac {\left (b^{3/2} c^3+12 \sqrt {-a} b d c^2+72 a \sqrt {b} d^2 c+64 (-a)^{3/2} d^3\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{32 a b^{5/2} d^{3/2}}-\frac {\left (b^{3/2} c^3+3 a \sqrt {b} d^2 c+2 \sqrt {-a} a d^3\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{4 a b^{5/2} d^{3/2}}+\frac {\left (b^{3/2} c^3-12 \sqrt {-a} b d c^2+72 a \sqrt {b} d^2 c+64 \sqrt {-a} a d^3\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{32 a b^{5/2} d^{3/2}}+\frac {\left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 \sqrt [4]{-a} b^{5/2} \sqrt {\sqrt {b} c+\sqrt {-a} d}}-\frac {\left (5 \sqrt {-a} b c^2-13 a \sqrt {b} d c-8 \sqrt {-a} a d^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{3/4} b^{5/2} \sqrt {\sqrt {b} c+\sqrt {-a} d}}+\frac {d (e x)^{5/2} \sqrt {c+d x}}{6 a b}+\frac {\left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{4 a b^2 d}+\frac {\left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{4 a b^2 d}+\frac {\left (b c^2-12 \sqrt {-a} \sqrt {b} d c+32 a d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{32 a b^2 d}+\frac {\left (b c^2+12 \sqrt {-a} \sqrt {b} d c+32 a d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{32 a b^2 d}\) |
Input:
Int[((e*x)^(5/2)*(c + d*x)^(3/2))/(a + b*x^2)^2,x]
Output:
(-7*c^2*e^2*Sqrt[e*x]*Sqrt[c + d*x])/(16*a*b*d) + ((b*c^2 - 2*Sqrt[-a]*Sqr t[b]*c*d - a*d^2)*e^2*Sqrt[e*x]*Sqrt[c + d*x])/(4*a*b^2*d) + ((b*c^2 + 2*S qrt[-a]*Sqrt[b]*c*d - a*d^2)*e^2*Sqrt[e*x]*Sqrt[c + d*x])/(4*a*b^2*d) + (( b*c^2 - 12*Sqrt[-a]*Sqrt[b]*c*d + 32*a*d^2)*e^2*Sqrt[e*x]*Sqrt[c + d*x])/( 32*a*b^2*d) + ((b*c^2 + 12*Sqrt[-a]*Sqrt[b]*c*d + 32*a*d^2)*e^2*Sqrt[e*x]* Sqrt[c + d*x])/(32*a*b^2*d) + (7*c*e*(e*x)^(3/2)*Sqrt[c + d*x])/(24*a*b) + (d*(e*x)^(5/2)*Sqrt[c + d*x])/(6*a*b) - ((Sqrt[b]*c - 8*Sqrt[-a]*d)*e^2*S qrt[e*x]*(c + d*x)^(3/2))/(16*a*b^(3/2)*d) - ((Sqrt[b]*c + 8*Sqrt[-a]*d)*e ^2*Sqrt[e*x]*(c + d*x)^(3/2))/(16*a*b^(3/2)*d) - (2*e*(e*x)^(3/2)*(c + d*x )^(3/2))/(3*a*b) - ((e*x)^(5/2)*(c + d*x)^(3/2))/(4*a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + ((e*x)^(5/2)*(c + d*x)^(3/2))/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[ b]*x)) + ((5*b*c^2 - 13*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*e^(5/2)*ArcTan[(Sq rt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])]) /(4*(-a)^(1/4)*b^(5/2)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]) - ((b*c^2 - 2*Sqrt[-a ]*Sqrt[b]*c*d - a*d^2)*e^(5/2)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e *x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(2*(-a)^(1/4)*b^(5/2)*Sqrt[Sqrt[ b]*c - Sqrt[-a]*d]) + (7*c^3*e^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]* Sqrt[c + d*x])])/(16*a*b*d^(3/2)) - ((Sqrt[b]*c + 2*Sqrt[-a]*d)*(b*c^2 - 2 *Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*e^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e ]*Sqrt[c + d*x])])/(4*a*b^(5/2)*d^(3/2)) + ((b^(3/2)*c^3 + 12*Sqrt[-a]*...
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3252\) vs. \(2(261)=522\).
Time = 0.51 (sec) , antiderivative size = 3253, normalized size of antiderivative = 9.32
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3253\) |
| risch | \(\text {Expression too large to display}\) | \(3559\) |
Input:
int((e*x)^(5/2)*(d*x+c)^(3/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/8*e^2*(e*x)^(1/2)*(d*x+c)^(1/2)*a*(-12*a^2*b*(-a*b)^(1/2)*((d*x+c)*e*x) ^(1/2)*(d*e)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*(e*(-a*d+c*(-a*b)^(1/ 2))/b)^(1/2)*d^3-8*a*b^2*(-a*b)^(1/2)*((d*x+c)*e*x)^(1/2)*(d*e)^(1/2)*(-e* (a*d+c*(-a*b)^(1/2))/b)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*d^3*x^2+9* a^2*b^2*(d*e)^(1/2)*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2 )*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)) )*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*c^3*d*e+3*a*b^2*(-a*b)^(1/2)*(d*e)^(1/ 2)*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a* b)^(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*(e*(-a*d+c*(-a* b)^(1/2))/b)^(1/2)*c^4*e+3*b^3*(-a*b)^(1/2)*(d*e)^(1/2)*ln((-2*(-a*b)^(1/2 )*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b- c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*c^ 4*e*x^2-9*a^3*b*(d*e)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*ln((2*(-a*b) ^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/ 2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*c*d^3*e-9*a^2*b^2*(d*e)^(1/2)*( -e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+ c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-( -a*b)^(1/2)))*c^3*d*e+3*a*b^2*(-a*b)^(1/2)*(d*e)^(1/2)*(-e*(a*d+c*(-a*b)^( 1/2))/b)^(1/2)*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*( -a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*c...
Leaf count of result is larger than twice the leaf count of optimal. 1256 vs. \(2 (261) = 522\).
Time = 0.30 (sec) , antiderivative size = 2511, normalized size of antiderivative = 7.19 \[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(5/2)*(d*x+c)^(3/2)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/8*(12*(b*c*e^2*x^2 + a*c*e^2)*sqrt(d*e)*log(2*d*e*x + c*e + 2*sqrt(d*e) *sqrt(d*x + c)*sqrt(e*x)) - 3*(b^3*x^2 + a*b^2)*sqrt(((5*b*c^2*d - 4*a*d^3 )*e^5 + b^5*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a*b^9) ))/b^5)*log(-27*((b^2*c^5 - 4*a*b*c^3*d^2 - 32*a^2*c*d^4)*sqrt(d*x + c)*sq rt(e*x)*e^7 + (2*a*b^7*d*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4) *e^10/(a*b^9))*x + (b^4*c^4 - 8*a*b^3*c^2*d^2)*e^5*x)*sqrt(((5*b*c^2*d - 4 *a*d^3)*e^5 + b^5*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/( a*b^9)))/b^5))/x) + 3*(b^3*x^2 + a*b^2)*sqrt(((5*b*c^2*d - 4*a*d^3)*e^5 + b^5*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a*b^9)))/b^5)* log(-27*((b^2*c^5 - 4*a*b*c^3*d^2 - 32*a^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x)* e^7 - (2*a*b^7*d*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a *b^9))*x + (b^4*c^4 - 8*a*b^3*c^2*d^2)*e^5*x)*sqrt(((5*b*c^2*d - 4*a*d^3)* e^5 + b^5*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a*b^9))) /b^5))/x) + 3*(b^3*x^2 + a*b^2)*sqrt(((5*b*c^2*d - 4*a*d^3)*e^5 - b^5*sqrt (-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a*b^9)))/b^5)*log(-27* ((b^2*c^5 - 4*a*b*c^3*d^2 - 32*a^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x)*e^7 + (2 *a*b^7*d*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a*b^9))*x - (b^4*c^4 - 8*a*b^3*c^2*d^2)*e^5*x)*sqrt(((5*b*c^2*d - 4*a*d^3)*e^5 - b^ 5*sqrt(-(b^2*c^6 - 16*a*b*c^4*d^2 + 64*a^2*c^2*d^4)*e^10/(a*b^9)))/b^5))/x ) - 3*(b^3*x^2 + a*b^2)*sqrt(((5*b*c^2*d - 4*a*d^3)*e^5 - b^5*sqrt(-(b^...
Timed out. \[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(5/2)*(d*x+c)**(3/2)/(b*x**2+a)**2,x)
Output:
Timed out
\[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^(5/2)*(d*x+c)^(3/2)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)*(e*x)^(5/2)/(b*x^2 + a)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 1292 vs. \(2 (261) = 522\).
Time = 1.37 (sec) , antiderivative size = 1292, normalized size of antiderivative = 3.70 \[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(5/2)*(d*x+c)^(3/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
-3/2*sqrt(d*e)*c*e^2*abs(d)*log((sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)* d*e - c*d*e))^2)/(b^2*d) + sqrt((d*x + c)*d*e - c*d*e)*sqrt(d*x + c)*e^2*a bs(d)/(b^2*d) - (sqrt(d*e)*b*c^5*d^3*e^6*abs(d) - 3*sqrt(d*e)*(sqrt(d*e)*s qrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*b*c^4*d^2*e^5*abs(d) - 2*sqr t(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*a*c^2*d^4 *e^5*abs(d) + 3*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b*c^3*d*e^4*abs(d) + 8*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt ((d*x + c)*d*e - c*d*e))^4*a*c*d^3*e^4*abs(d) - sqrt(d*e)*(sqrt(d*e)*sqrt( d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^6*b*c^2*e^3*abs(d) + 2*sqrt(d*e)*( sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^6*a*d^2*e^3*abs(d)) /((b*c^4*d^4*e^4 - 4*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e ))^2*b*c^3*d^3*e^3 + 6*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d *e))^4*b*c^2*d^2*e^2 + 16*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a*d^4*e^2 - 4*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c* d*e))^6*b*c*d*e + (sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^ 8*b)*b^2) + 3/8*(sqrt(d*e)*b^2*c^6*d^4*e^5*abs(d) - 8*sqrt(d*e)*a*b*c^4*d^ 6*e^5*abs(d) - 4*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b^2*c^4*d^2*e^3*abs(d) - 44*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a*b*c^2*d^4*e^3*abs(d) - 64*sqrt(d*e)*(sq rt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a^2*d^6*e^3*abs(...
Timed out. \[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int(((e*x)^(5/2)*(c + d*x)^(3/2))/(a + b*x^2)^2,x)
Output:
int(((e*x)^(5/2)*(c + d*x)^(3/2))/(a + b*x^2)^2, x)
\[ \int \frac {(e x)^{5/2} (c+d x)^{3/2}}{\left (a+b x^2\right )^2} \, dx=\sqrt {e}\, e^{2} \left (\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, x^{3}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) d +\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, x^{2}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) c \right ) \] Input:
int((e*x)^(5/2)*(d*x+c)^(3/2)/(b*x^2+a)^2,x)
Output:
sqrt(e)*e**2*(int((sqrt(x)*sqrt(c + d*x)*x**3)/(a**2 + 2*a*b*x**2 + b**2*x **4),x)*d + int((sqrt(x)*sqrt(c + d*x)*x**2)/(a**2 + 2*a*b*x**2 + b**2*x** 4),x)*c)