Integrand size = 28, antiderivative size = 275 \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}+\frac {2 e^{-\frac {2 a}{b p q}} h \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
2*(-e*h+f*g)*(f*x+e)*erfi((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2 )/q^(1/2))*Pi^(1/2)/b^(3/2)/exp(a/b/p/q)/f^2/p^(3/2)/q^(3/2)/((c*(d*(f*x+e )^p)^q)^(1/p/q))+2*h*(f*x+e)^2*erfi(2^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1 /2)/b^(1/2)/p^(1/2)/q^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/exp(2*a/b/p/q)/f^2/p ^(3/2)/q^(3/2)/((c*(d*(f*x+e)^p)^q)^(2/p/q))-2*(f*x+e)*(h*x+g)/b/f/p/q/(a+ b*ln(c*(d*(f*x+e)^p)^q))^(1/2)
Time = 0.73 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.58 \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {2 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (-2 e e^{\frac {a}{b p q}} h \sqrt {\pi } \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}+h \sqrt {2 \pi } (e+f x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}+\sqrt {b} e^{\frac {a}{b p q}} \sqrt {p} \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \left (-e^{\frac {a}{b p q}} f \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} (g+h x)+(f g+e h) \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}\right )\right )}{b^{3/2} f^2 p^{3/2} q^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
(2*(e + f*x)*(-2*e*E^(a/(b*p*q))*h*Sqrt[Pi]*(c*(d*(e + f*x)^p)^q)^(1/(p*q) )*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqr t[a + b*Log[c*(d*(e + f*x)^p)^q]] + h*Sqrt[2*Pi]*(e + f*x)*Erfi[(Sqrt[2]*S qrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b *Log[c*(d*(e + f*x)^p)^q]] + Sqrt[b]*E^(a/(b*p*q))*Sqrt[p]*Sqrt[q]*(c*(d*( e + f*x)^p)^q)^(1/(p*q))*(-(E^(a/(b*p*q))*f*(c*(d*(e + f*x)^p)^q)^(1/(p*q) )*(g + h*x)) + (f*g + e*h)*Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^p)^q])/( b*p*q))]*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))])))/(b^(3/2)*E^( (2*a)/(b*p*q))*f^2*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])
Time = 1.82 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.46, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2895, 2847, 2836, 2737, 2611, 2633, 2848, 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.
| \(\displaystyle \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {2 (f g-e h) \int \frac {1}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle -\frac {2 (f g-e h) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}d(e+f x)}{b f^2 p q}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle -\frac {2 (e+f x) (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \frac {\left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}}}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}d\log \left (c d^q (e+f x)^{p q}\right )}{b f^2 p^2 q^2}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 (e+f x) (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \exp \left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}-\frac {a}{b p q}\right )d\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{b^2 f^2 p^2 q^2}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \frac {4 \int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {h (e+f x)}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )dx}{b p q}-\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}+\frac {4 \left (\frac {\sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {\sqrt {\frac {\pi }{2}} h (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\) |
(-2*(f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*d^q*(e + f*x)^(p* q)]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^(a/(b*p*q))*f^2*p^(3/2)*q^(3/2 )*(c*d^q*(e + f*x)^(p*q))^(1/(p*q))) + (4*(((f*g - e*h)*Sqrt[Pi]*(e + f*x) *Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sq rt[b]*E^(a/(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*Sqrt[Pi/2]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^ q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/(b*p*q))*f^2*Sqrt[p]*Sq rt[q]*(c*(d*(e + f*x)^p)^q)^(2/(p*q)))))/(b*p*q) - (2*(e + f*x)*(g + h*x)) /(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])
3.5.75.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x ^n)^(1/n)) Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ [{a, b, c, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e *x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1)) Int[( f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) /(b*e*n*(p + 1))) Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 ), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt Q[p, -1] && GtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
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]]
\[\int \frac {h x +g}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\int \frac {g + h x}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\int { \frac {h x + g}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\int { \frac {h x + g}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\int \frac {g+h\,x}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{3/2}} \,d x \]