\(\int \frac {x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) [902]

Optimal result

Integrand size = 20, antiderivative size = 130 \[ \int \frac {x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d}-\frac {(b c+3 a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac {(b c+3 a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}} \]

[Out]

(b*x+a)^(1/4)*(d*x+c)^(3/4)/b/d-1/2*(3*a*d+b*c)*arctan(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(7/4)/d^
(5/4)-1/2*(3*a*d+b*c)*arctanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(7/4)/d^(5/4)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {81, 65, 246, 218, 214, 211} \[ \int \frac {x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {(3 a d+b c) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac {(3 a d+b c) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

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

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]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96 \[ \int \frac {x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {2 b^{3/4} \sqrt [4]{d} \sqrt [4]{a+b x} (c+d x)^{3/4}-(b c+3 a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )-(b c+3 a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

int(x/(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 = 715, normalized size of antiderivative = 5.50 \[ \int \frac {x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - i \, b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (i \, b^{2} d^{2} x + i \, b^{2} c d\right )} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + i \, b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (-i \, b^{2} d^{2} x - i \, b^{2} c d\right )} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - 4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, b d} \]

[In]

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

[Out]

-1/4*(b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*log
(((b*c + 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^2*d^2*x + b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^
2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4))/(d*x + c)) - b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a
^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*log(((b*c + 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(
3/4) - (b^2*d^2*x + b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(
b^7*d^5))^(1/4))/(d*x + c)) - I*b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4
*d^4)/(b^7*d^5))^(1/4)*log(((b*c + 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (I*b^2*d^2*x + I*b^2*c*d)*((b^4*c^
4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4))/(d*x + c)) + I*b*d*(
(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*log(((b*c + 3*
a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (-I*b^2*d^2*x - I*b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*
d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4))/(d*x + c)) - 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(b*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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