Integrand size = 29, antiderivative size = 481 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {8 b e f p}{21 d h^3 (h x)^{3/2}}-\frac {8 b e g p}{5 d h^4 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{7/4} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{5/4} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} f p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} g p \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}}{\sqrt {h} \left (\sqrt {d}+\sqrt {e} x\right )}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}} \] Output:
-8/21*b*e*f*p/d/h^3/(h*x)^(3/2)-8/5*b*e*g*p/d/h^4/(h*x)^(1/2)+2/7*2^(1/2)* b*e^(7/4)*f*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(7/4 )/h^(9/2)+2/5*2^(1/2)*b*e^(5/4)*g*p*arctan(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d ^(1/4)/h^(1/2))/d^(5/4)/h^(9/2)-2/7*2^(1/2)*b*e^(7/4)*f*p*arctan(1+2^(1/2) *e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(7/4)/h^(9/2)-2/5*2^(1/2)*b*e^(5/4 )*g*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/d^(5/4)/h^(9/2 )-2/7*2^(1/2)*b*e^(7/4)*f*p*arctanh(2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1/2)/h^ (1/2)/(d^(1/2)+e^(1/2)*x))/d^(7/4)/h^(9/2)+2/5*2^(1/2)*b*e^(5/4)*g*p*arcta nh(2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1/2)/h^(1/2)/(d^(1/2)+e^(1/2)*x))/d^(5/4 )/h^(9/2)-2/7*f*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(7/2)-2/5*g*(a+b*ln(c*(e*x ^2+d)^p))/h^2/(h*x)^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.21 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {2 \sqrt {h x} \left (20 b e f p x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {e x^2}{d}\right )+84 b e g p x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )+3 d (5 f+7 g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{105 d h^5 x^4} \] Input:
Integrate[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(9/2),x]
Output:
(-2*Sqrt[h*x]*(20*b*e*f*p*x^2*Hypergeometric2F1[-3/4, 1, 1/4, -((e*x^2)/d) ] + 84*b*e*g*p*x^3*Hypergeometric2F1[-1/4, 1, 3/4, -((e*x^2)/d)] + 3*d*(5* f + 7*g*x)*(a + b*Log[c*(d + e*x^2)^p])))/(105*d*h^5*x^4)
Time = 1.56 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2917, 27, 2926, 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.
| \(\displaystyle \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2917 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^5 x^4}d\sqrt {h x}}{h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {(f h+g x h) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^4 x^4}d\sqrt {h x}}{h^2}\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \frac {2 \int \left (\frac {g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 x^3}+\frac {f \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 x^4}\right )d\sqrt {h x}}{h^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {f h \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 (h x)^{7/2}}-\frac {g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 (h x)^{5/2}}+\frac {\sqrt {2} b e^{7/4} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{5/2}}-\frac {\sqrt {2} b e^{7/4} f p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{7 d^{7/4} h^{5/2}}+\frac {\sqrt {2} b e^{5/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{5/2}}-\frac {\sqrt {2} b e^{5/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{5 d^{5/4} h^{5/2}}+\frac {b e^{7/4} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{7 \sqrt {2} d^{7/4} h^{5/2}}-\frac {b e^{7/4} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{7 \sqrt {2} d^{7/4} h^{5/2}}-\frac {b e^{5/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{5 \sqrt {2} d^{5/4} h^{5/2}}+\frac {b e^{5/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h} \sqrt {h x}+\sqrt {d} h+\sqrt {e} h x\right )}{5 \sqrt {2} d^{5/4} h^{5/2}}-\frac {4 b e f p}{21 d h (h x)^{3/2}}-\frac {4 b e g p}{5 d h^2 \sqrt {h x}}\right )}{h^2}\) |
Input:
Int[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(9/2),x]
Output:
(2*((-4*b*e*f*p)/(21*d*h*(h*x)^(3/2)) - (4*b*e*g*p)/(5*d*h^2*Sqrt[h*x]) + (Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqr t[h])])/(7*d^(7/4)*h^(5/2)) + (Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 - (Sqrt[2]*e ^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(5/2)) - (Sqrt[2]*b*e^( 7/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^( 7/4)*h^(5/2)) - (Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h* x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(5/2)) - (f*h*(a + b*Log[c*(d + e*x^2 )^p]))/(7*(h*x)^(7/2)) - (g*(a + b*Log[c*(d + e*x^2)^p]))/(5*(h*x)^(5/2)) + (b*e^(7/4)*f*p*Log[Sqrt[d]*h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqr t[h]*Sqrt[h*x]])/(7*Sqrt[2]*d^(7/4)*h^(5/2)) - (b*e^(5/4)*g*p*Log[Sqrt[d]* h + Sqrt[e]*h*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(5*Sqrt[2]*d ^(5/4)*h^(5/2)) - (b*e^(7/4)*f*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^( 1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x]])/(7*Sqrt[2]*d^(7/4)*h^(5/2)) + (b*e^(5/4)* g*p*Log[Sqrt[d]*h + Sqrt[e]*h*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h]*Sqrt[h*x ]])/(5*Sqrt[2]*d^(5/4)*h^(5/2))))/h^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.) *(x_))^(m_)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> With[{k = Denominator[ m]}, Simp[k/h Subst[Int[x^(k*(m + 1) - 1)*(f + g*(x^k/h))^r*(a + b*Log[c* (d + e*(x^(k*n)/h^n))^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b *Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e , f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & & IntegerQ[s]
\[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {9}{2}}}d x\]
Input:
int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(9/2),x)
Output:
int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(9/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 1369 vs. \(2 (337) = 674\).
Time = 0.15 (sec) , antiderivative size = 1369, normalized size of antiderivative = 2.85 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x, algorithm="frica s")
Output:
2/105*(3*d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6* f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3 *h^9))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 + 32 *(7*d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4* d^2*e^5*g^4)*p^4/(d^7*h^18)) + 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^ 2)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - 3*d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2* g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9 ))*log(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 - 32*(7* d^6*g*h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2* e^5*g^4)*p^4/(d^7*h^18)) + 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h ^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2 401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) + 70*b^2*e^3*f*g*p^2)/(d^3*h^9))) - 3 *d*h^5*x^4*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g^4)*p^4/(d^7*h^18)) - 70*b^2*e^3*f*g*p^2)/(d^3*h^9))*lo g(-32*(625*b^3*e^6*f^4 - 2401*b^3*d^2*e^4*g^4)*sqrt(h*x)*p^3 + 32*(7*d^6*g *h^14*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401*b^4*d^2*e^5*g ^4)*p^4/(d^7*h^18)) - 5*(25*b^2*d^2*e^4*f^3 - 49*b^2*d^3*e^3*f*g^2)*h^5*p^ 2)*sqrt((d^3*h^9*sqrt(-(625*b^4*e^7*f^4 - 2450*b^4*d*e^6*f^2*g^2 + 2401...
Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(9/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (337) = 674\).
Time = 0.14 (sec) , antiderivative size = 739, normalized size of antiderivative = 1.54 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx =\text {Too large to display} \] Input:
integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x, algorithm="maxim a")
Output:
-1/21*b*e*f*p*(3*(sqrt(2)*e^(3/4)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)* sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h^2)^(3/4) - sqrt(2)*e^(3/4)*log(sqrt(e) *h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h^2)^(3/4) + sqrt(2)*e*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4) *e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2 )*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)* sqrt(d)*h) + sqrt(2)*e*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d *h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)* h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)* sqrt(e)*h)*sqrt(d)*h))/d + 8/((h*x)^(3/2)*d))/h^3 + 1/5*b*e*g*p*(e*(sqrt(2 )*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/( (d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*s qrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(-(sqrt (2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x) *sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4 ) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)) - sqrt(2)*log (-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sq rt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4) *e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)))/d - 8 /(sqrt(h*x)*d))/h^4 - 2/5*b*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(9/2) - 2/...
Time = 0.17 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.05 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {\frac {6 \, {\left (7 \, b g h p x + 5 \, b f h p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{4} x^{3}} + \frac {6 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p + 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e h^{5}} + \frac {6 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p + 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e h^{5}} + \frac {3 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p - 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e h^{5}} - \frac {3 \, {\left (5 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f h p - 7 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b g p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e h^{5}} + \frac {2 \, {\left (84 \, b e g h^{3} p x^{3} + 20 \, b e f h^{3} p x^{2} - 21 \, b d g h^{3} p x \log \left (h^{2}\right ) - 15 \, b d f h^{3} p \log \left (h^{2}\right ) + 21 \, b d g h^{3} x \log \left (c\right ) + 21 \, a d g h^{3} x + 15 \, b d f h^{3} \log \left (c\right ) + 15 \, a d f h^{3}\right )}}{\sqrt {h x} d h^{6} x^{3}}}{105 \, h} \] Input:
integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x, algorithm="giac" )
Output:
-1/105*(6*(7*b*g*h*p*x + 5*b*f*h*p)*log(e*h^2*x^2 + d*h^2)/(sqrt(h*x)*h^4* x^3) + 6*(5*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f*h*p + 7*sqrt(2)*(d*e^3*h^2)^ (3/4)*b*g*p)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) + 2*sqrt(h*x))/(d *h^2/e)^(1/4))/(d^2*e*h^5) + 6*(5*sqrt(2)*(d*e^3*h^2)^(1/4)*b*e^2*f*h*p + 7*sqrt(2)*(d*e^3*h^2)^(3/4)*b*g*p)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^ (1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/(d^2*e*h^5) + 3*(5*sqrt(2)*(d*e^3*h^ 2)^(1/4)*b*e^2*f*h*p - 7*sqrt(2)*(d*e^3*h^2)^(3/4)*b*g*p)*log(h*x + sqrt(2 )*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d^2*e*h^5) - 3*(5*sqrt(2)*(d *e^3*h^2)^(1/4)*b*e^2*f*h*p - 7*sqrt(2)*(d*e^3*h^2)^(3/4)*b*g*p)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d^2*e*h^5) + 2*(84*b* e*g*h^3*p*x^3 + 20*b*e*f*h^3*p*x^2 - 21*b*d*g*h^3*p*x*log(h^2) - 15*b*d*f* h^3*p*log(h^2) + 21*b*d*g*h^3*x*log(c) + 21*a*d*g*h^3*x + 15*b*d*f*h^3*log (c) + 15*a*d*f*h^3)/(sqrt(h*x)*d*h^6*x^3))/h
Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{9/2}} \,d x \] Input:
int(((f + g*x)*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(9/2),x)
Output:
int(((f + g*x)*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(9/2), x)
Time = 0.17 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\frac {\sqrt {h}\, \left (42 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p \,x^{3}+30 \sqrt {x}\, e^{\frac {7}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p \,x^{3}-42 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p \,x^{3}-30 \sqrt {x}\, e^{\frac {7}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b f p \,x^{3}-42 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b g p \,x^{3}+21 \sqrt {x}\, e^{\frac {5}{4}} d^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b g \,x^{3}+30 \sqrt {x}\, e^{\frac {7}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b f p \,x^{3}-15 \sqrt {x}\, e^{\frac {7}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b f \,x^{3}-30 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b \,d^{2} f -42 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) b \,d^{2} g x -30 a \,d^{2} f -42 a \,d^{2} g x -40 b d e f p \,x^{2}-168 b d e g p \,x^{3}\right )}{105 \sqrt {x}\, d^{2} h^{5} x^{3}} \] Input:
int((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(9/2),x)
Output:
(sqrt(h)*(42*sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqr t(2) - 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*e*g*p*x**3 + 30*s qrt(x)*e**(3/4)*d**(1/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) - 2*sqrt( x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*e*f*p*x**3 - 42*sqrt(x)*e**(1/4 )*d**(3/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(e))/(e **(1/4)*d**(1/4)*sqrt(2)))*b*e*g*p*x**3 - 30*sqrt(x)*e**(3/4)*d**(1/4)*sqr t(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/ 4)*sqrt(2)))*b*e*f*p*x**3 - 42*sqrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*log( - sq rt(x)*e**(1/4)*d**(1/4)*sqrt(2) + sqrt(d) + sqrt(e)*x)*b*e*g*p*x**3 + 21*s qrt(x)*e**(1/4)*d**(3/4)*sqrt(2)*log((d + e*x**2)**p*c)*b*e*g*x**3 + 30*sq rt(x)*e**(3/4)*d**(1/4)*sqrt(2)*log( - sqrt(x)*e**(1/4)*d**(1/4)*sqrt(2) + sqrt(d) + sqrt(e)*x)*b*e*f*p*x**3 - 15*sqrt(x)*e**(3/4)*d**(1/4)*sqrt(2)* log((d + e*x**2)**p*c)*b*e*f*x**3 - 30*log((d + e*x**2)**p*c)*b*d**2*f - 4 2*log((d + e*x**2)**p*c)*b*d**2*g*x - 30*a*d**2*f - 42*a*d**2*g*x - 40*b*d *e*f*p*x**2 - 168*b*d*e*g*p*x**3))/(105*sqrt(x)*d**2*h**5*x**3)