\(\int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) [906]

Optimal result

Integrand size = 22, antiderivative size = 206 \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}} \]

[Out]

-1/2*(b*x+a)^(1/4)*(d*x+c)^(3/4)/a/c/x^2+1/8*(5*a*d+7*b*c)*(b*x+a)^(1/4)*(d*x+c)^(3/4)/a^2/c^2/x-1/16*(5*a^2*d
^2+6*a*b*c*d+21*b^2*c^2)*arctan(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(11/4)/c^(9/4)-1/16*(5*a^2*d^2+
6*a*b*c*d+21*b^2*c^2)*arctanh(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(11/4)/c^(9/4)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {105, 156, 12, 95, 218, 214, 211} \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{8 a^2 c^2 x}-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2} \]

[In]

Int[1/(x^3*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

-1/2*((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x^2) + ((7*b*c + 5*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*a^2*c^
2*x) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*
a^(11/4)*c^(9/4)) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)
^(1/4))])/(16*a^(11/4)*c^(9/4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}-\frac {\int \frac {\frac {1}{4} (7 b c+5 a d)+b d x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2 a c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}+\frac {\int \frac {21 b^2 c^2+6 a b c d+5 a^2 d^2}{16 x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2 a^2 c^2} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 a^2 c^2} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{8 a^2 c^2} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {c} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 a^{5/2} c^2}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {c} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 a^{5/2} c^2} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4} (-4 a c+7 b c x+5 a d x)-\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) x^2 \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )-\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) x^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4} x^2} \]

[In]

Integrate[1/(x^3*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

(2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4)*(-4*a*c + 7*b*c*x + 5*a*d*x) - (21*b^2*c^2 + 6*a*b*c*d + 5*
a^2*d^2)*x^2*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))] - (21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2
)*x^2*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*a^(11/4)*c^(9/4)*x^2)

Maple [F]

\[\int \frac {1}{x^{3} \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]

[In]

int(1/x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(1/x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 1287, normalized size of antiderivative = 6.25 \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

-1/32*(a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 1
12806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*
c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^3*c^2*d*x + a^3*c^3
)*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4
*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(1/4))/(d*x
+ c)) - a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 +
112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^1
1*c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^3*c^2*d*x + a^3*c
^3)*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c
^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(1/4))/(d
*x + c)) - I*a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d
^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/
(a^11*c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (I*a^3*c^2*d*x +
I*a^3*c^3)*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a
^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(
1/4))/(d*x + c)) + I*a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b
^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a
^8*d^8)/(a^11*c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (-I*a^3*
c^2*d*x - I*a^3*c^3)*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 +
112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^1
1*c^9))^(1/4))/(d*x + c)) + 4*(4*a*c - (7*b*c + 5*a*d)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a^2*c^2*x^2)

Sympy [F]

\[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^{3} \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]

[In]

integrate(1/x**3/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(1/(x**3*(a + b*x)**(3/4)*(c + d*x)**(1/4)), x)

Maxima [F]

\[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x^3), x)

Giac [F]

\[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^3\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]

[In]

int(1/(x^3*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x)

[Out]

int(1/(x^3*(a + b*x)^(3/4)*(c + d*x)^(1/4)), x)