\(\int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx\) [1706]

Optimal result

Integrand size = 19, antiderivative size = 127 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}+\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {52, 65, 338, 304, 211, 214} \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}+\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d} \]

[In]

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

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 \sqrt [4]{b} d^{7/4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 691, normalized size of antiderivative = 5.44 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=-\frac {3 \, d \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (b d^{2} x + a d^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 3 \, d \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} - {\left (b d^{2} x + a d^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 3 i \, d \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (i \, b d^{2} x + i \, a d^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 3 i \, d \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (-i \, b d^{2} x - i \, a d^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b d^{7}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 4 \, {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{4 \, d} \]

[In]

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

[Out]

-1/4*(3*d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4)*log(-3*((b*c
- a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b*d^2*x + a*d^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*
a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) - 3*d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*
b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4)*log(-3*((b*c - a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b*d^2*x + a*d^2)*((b^
4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) + 3*I*d*((b^4*
c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4)*log(-3*((b*c - a*d)*(b*x + a
)^(3/4)*(d*x + c)^(1/4) + (I*b*d^2*x + I*a*d^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3
+ a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) - 3*I*d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)/(b*d^7))^(1/4)*log(-3*((b*c - a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (-I*b*d^2*x - I*a*d^2)*((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) - 4*(b*x + a)^(3/4)
*(d*x + c)^(1/4))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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