Integrand size = 28, antiderivative size = 152 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\frac {4 b f p q}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {4 b f^2 p q}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {4 b f^{5/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{5 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}} \]
4/15*b*f*p*q/h/(-e*h+f*g)/(h*x+g)^(3/2)-4/5*b*f^(5/2)*p*q*arctanh(f^(1/2)* (h*x+g)^(1/2)/(-e*h+f*g)^(1/2))/h/(-e*h+f*g)^(5/2)-2/5*(a+b*ln(c*(d*(f*x+e )^p)^q))/h/(h*x+g)^(5/2)+4/5*b*f^2*p*q/h/(-e*h+f*g)^2/(h*x+g)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.60 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\frac {-4 b f p q (g+h x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {f (g+h x)}{f g-e h}\right )+6 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (-f g+e h) (g+h x)^{5/2}} \]
(-4*b*f*p*q*(g + h*x)*Hypergeometric2F1[-3/2, 1, -1/2, (f*(g + h*x))/(f*g - e*h)] + 6*(f*g - e*h)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(15*h*(-(f*g) + e*h)*(g + h*x)^(5/2))
Time = 0.44 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2895, 2842, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}}dx\) |
| \(\Big \downarrow \) 2842 |
| \(\displaystyle \frac {2 b f p q \int \frac {1}{(e+f x) (g+h x)^{5/2}}dx}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {2 b f p q \left (\frac {f \int \frac {1}{(e+f x) (g+h x)^{3/2}}dx}{f g-e h}+\frac {2}{3 (g+h x)^{3/2} (f g-e h)}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {2 b f p q \left (\frac {f \left (\frac {f \int \frac {1}{(e+f x) \sqrt {g+h x}}dx}{f g-e h}+\frac {2}{\sqrt {g+h x} (f g-e h)}\right )}{f g-e h}+\frac {2}{3 (g+h x)^{3/2} (f g-e h)}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 b f p q \left (\frac {f \left (\frac {2 f \int \frac {1}{e+\frac {f (g+h x)}{h}-\frac {f g}{h}}d\sqrt {g+h x}}{h (f g-e h)}+\frac {2}{\sqrt {g+h x} (f g-e h)}\right )}{f g-e h}+\frac {2}{3 (g+h x)^{3/2} (f g-e h)}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 b f p q \left (\frac {f \left (\frac {2}{\sqrt {g+h x} (f g-e h)}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{f g-e h}+\frac {2}{3 (g+h x)^{3/2} (f g-e h)}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (g+h x)^{5/2}}\) |
(2*b*f*p*q*(2/(3*(f*g - e*h)*(g + h*x)^(3/2)) + (f*(2/((f*g - e*h)*Sqrt[g + h*x]) - (2*Sqrt[f]*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(f* g - e*h)^(3/2)))/(f*g - e*h)))/(5*h) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]) )/(5*h*(g + h*x)^(5/2))
3.5.87.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right )^{\frac {7}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (128) = 256\).
Time = 0.37 (sec) , antiderivative size = 863, normalized size of antiderivative = 5.68 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\left [\frac {2 \, {\left (3 \, {\left (b f^{2} h^{3} p q x^{3} + 3 \, b f^{2} g h^{2} p q x^{2} + 3 \, b f^{2} g^{2} h p q x + b f^{2} g^{3} p q\right )} \sqrt {\frac {f}{f g - e h}} \log \left (\frac {f h x + 2 \, f g - e h - 2 \, {\left (f g - e h\right )} \sqrt {h x + g} \sqrt {\frac {f}{f g - e h}}}{f x + e}\right ) + {\left (6 \, b f^{2} h^{2} p q x^{2} - 3 \, a f^{2} g^{2} + 6 \, a e f g h - 3 \, a e^{2} h^{2} + 2 \, {\left (7 \, b f^{2} g h - b e f h^{2}\right )} p q x - 3 \, {\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} p q \log \left (f x + e\right ) + 2 \, {\left (4 \, b f^{2} g^{2} - b e f g h\right )} p q - 3 \, {\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} q \log \left (d\right ) - 3 \, {\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} \log \left (c\right )\right )} \sqrt {h x + g}\right )}}{15 \, {\left (f^{2} g^{5} h - 2 \, e f g^{4} h^{2} + e^{2} g^{3} h^{3} + {\left (f^{2} g^{2} h^{4} - 2 \, e f g h^{5} + e^{2} h^{6}\right )} x^{3} + 3 \, {\left (f^{2} g^{3} h^{3} - 2 \, e f g^{2} h^{4} + e^{2} g h^{5}\right )} x^{2} + 3 \, {\left (f^{2} g^{4} h^{2} - 2 \, e f g^{3} h^{3} + e^{2} g^{2} h^{4}\right )} x\right )}}, -\frac {2 \, {\left (6 \, {\left (b f^{2} h^{3} p q x^{3} + 3 \, b f^{2} g h^{2} p q x^{2} + 3 \, b f^{2} g^{2} h p q x + b f^{2} g^{3} p q\right )} \sqrt {-\frac {f}{f g - e h}} \arctan \left (-\frac {{\left (f g - e h\right )} \sqrt {h x + g} \sqrt {-\frac {f}{f g - e h}}}{f h x + f g}\right ) - {\left (6 \, b f^{2} h^{2} p q x^{2} - 3 \, a f^{2} g^{2} + 6 \, a e f g h - 3 \, a e^{2} h^{2} + 2 \, {\left (7 \, b f^{2} g h - b e f h^{2}\right )} p q x - 3 \, {\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} p q \log \left (f x + e\right ) + 2 \, {\left (4 \, b f^{2} g^{2} - b e f g h\right )} p q - 3 \, {\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} q \log \left (d\right ) - 3 \, {\left (b f^{2} g^{2} - 2 \, b e f g h + b e^{2} h^{2}\right )} \log \left (c\right )\right )} \sqrt {h x + g}\right )}}{15 \, {\left (f^{2} g^{5} h - 2 \, e f g^{4} h^{2} + e^{2} g^{3} h^{3} + {\left (f^{2} g^{2} h^{4} - 2 \, e f g h^{5} + e^{2} h^{6}\right )} x^{3} + 3 \, {\left (f^{2} g^{3} h^{3} - 2 \, e f g^{2} h^{4} + e^{2} g h^{5}\right )} x^{2} + 3 \, {\left (f^{2} g^{4} h^{2} - 2 \, e f g^{3} h^{3} + e^{2} g^{2} h^{4}\right )} x\right )}}\right ] \]
[2/15*(3*(b*f^2*h^3*p*q*x^3 + 3*b*f^2*g*h^2*p*q*x^2 + 3*b*f^2*g^2*h*p*q*x + b*f^2*g^3*p*q)*sqrt(f/(f*g - e*h))*log((f*h*x + 2*f*g - e*h - 2*(f*g - e *h)*sqrt(h*x + g)*sqrt(f/(f*g - e*h)))/(f*x + e)) + (6*b*f^2*h^2*p*q*x^2 - 3*a*f^2*g^2 + 6*a*e*f*g*h - 3*a*e^2*h^2 + 2*(7*b*f^2*g*h - b*e*f*h^2)*p*q *x - 3*(b*f^2*g^2 - 2*b*e*f*g*h + b*e^2*h^2)*p*q*log(f*x + e) + 2*(4*b*f^2 *g^2 - b*e*f*g*h)*p*q - 3*(b*f^2*g^2 - 2*b*e*f*g*h + b*e^2*h^2)*q*log(d) - 3*(b*f^2*g^2 - 2*b*e*f*g*h + b*e^2*h^2)*log(c))*sqrt(h*x + g))/(f^2*g^5*h - 2*e*f*g^4*h^2 + e^2*g^3*h^3 + (f^2*g^2*h^4 - 2*e*f*g*h^5 + e^2*h^6)*x^3 + 3*(f^2*g^3*h^3 - 2*e*f*g^2*h^4 + e^2*g*h^5)*x^2 + 3*(f^2*g^4*h^2 - 2*e* f*g^3*h^3 + e^2*g^2*h^4)*x), -2/15*(6*(b*f^2*h^3*p*q*x^3 + 3*b*f^2*g*h^2*p *q*x^2 + 3*b*f^2*g^2*h*p*q*x + b*f^2*g^3*p*q)*sqrt(-f/(f*g - e*h))*arctan( -(f*g - e*h)*sqrt(h*x + g)*sqrt(-f/(f*g - e*h))/(f*h*x + f*g)) - (6*b*f^2* h^2*p*q*x^2 - 3*a*f^2*g^2 + 6*a*e*f*g*h - 3*a*e^2*h^2 + 2*(7*b*f^2*g*h - b *e*f*h^2)*p*q*x - 3*(b*f^2*g^2 - 2*b*e*f*g*h + b*e^2*h^2)*p*q*log(f*x + e) + 2*(4*b*f^2*g^2 - b*e*f*g*h)*p*q - 3*(b*f^2*g^2 - 2*b*e*f*g*h + b*e^2*h^ 2)*q*log(d) - 3*(b*f^2*g^2 - 2*b*e*f*g*h + b*e^2*h^2)*log(c))*sqrt(h*x + g ))/(f^2*g^5*h - 2*e*f*g^4*h^2 + e^2*g^3*h^3 + (f^2*g^2*h^4 - 2*e*f*g*h^5 + e^2*h^6)*x^3 + 3*(f^2*g^3*h^3 - 2*e*f*g^2*h^4 + e^2*g*h^5)*x^2 + 3*(f^2*g ^4*h^2 - 2*e*f*g^3*h^3 + e^2*g^2*h^4)*x)]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*h-f*g>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (128) = 256\).
Time = 0.34 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\frac {4 \, b f^{3} h p q \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + e f h}}\right )}{5 \, {\left (f^{2} g^{2} h^{2} - 2 \, e f g h^{3} + e^{2} h^{4}\right )} \sqrt {-f^{2} g + e f h}} - \frac {2 \, b p q \log \left ({\left (h x + g\right )} f - f g + e h\right )}{5 \, {\left (h x + g\right )}^{\frac {5}{2}} h} + \frac {2 \, {\left (3 \, b f^{2} g^{2} p q \log \left (h\right ) - 6 \, b e f g h p q \log \left (h\right ) + 3 \, b e^{2} h^{2} p q \log \left (h\right ) + 6 \, {\left (h x + g\right )}^{2} b f^{2} p q + 2 \, {\left (h x + g\right )} b f^{2} g p q - 2 \, {\left (h x + g\right )} b e f h p q - 3 \, b f^{2} g^{2} q \log \left (d\right ) + 6 \, b e f g h q \log \left (d\right ) - 3 \, b e^{2} h^{2} q \log \left (d\right ) - 3 \, b f^{2} g^{2} \log \left (c\right ) + 6 \, b e f g h \log \left (c\right ) - 3 \, b e^{2} h^{2} \log \left (c\right ) - 3 \, a f^{2} g^{2} + 6 \, a e f g h - 3 \, a e^{2} h^{2}\right )}}{15 \, {\left ({\left (h x + g\right )}^{\frac {5}{2}} f^{2} g^{2} h - 2 \, {\left (h x + g\right )}^{\frac {5}{2}} e f g h^{2} + {\left (h x + g\right )}^{\frac {5}{2}} e^{2} h^{3}\right )}} \]
4/5*b*f^3*h*p*q*arctan(sqrt(h*x + g)*f/sqrt(-f^2*g + e*f*h))/((f^2*g^2*h^2 - 2*e*f*g*h^3 + e^2*h^4)*sqrt(-f^2*g + e*f*h)) - 2/5*b*p*q*log((h*x + g)* f - f*g + e*h)/((h*x + g)^(5/2)*h) + 2/15*(3*b*f^2*g^2*p*q*log(h) - 6*b*e* f*g*h*p*q*log(h) + 3*b*e^2*h^2*p*q*log(h) + 6*(h*x + g)^2*b*f^2*p*q + 2*(h *x + g)*b*f^2*g*p*q - 2*(h*x + g)*b*e*f*h*p*q - 3*b*f^2*g^2*q*log(d) + 6*b *e*f*g*h*q*log(d) - 3*b*e^2*h^2*q*log(d) - 3*b*f^2*g^2*log(c) + 6*b*e*f*g* h*log(c) - 3*b*e^2*h^2*log(c) - 3*a*f^2*g^2 + 6*a*e*f*g*h - 3*a*e^2*h^2)/( (h*x + g)^(5/2)*f^2*g^2*h - 2*(h*x + g)^(5/2)*e*f*g*h^2 + (h*x + g)^(5/2)* e^2*h^3)
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{7/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{{\left (g+h\,x\right )}^{7/2}} \,d x \]