\(\int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx\) [1696]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=-\frac {4 (c+d x)^{3/4}}{3 (b c-a d) (a+b x)^{3/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=-\frac {4 (c+d x)^{3/4}}{3 (a+b x)^{3/4} (b c-a d)} \]

[In]

Int[1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x]

[Out]

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

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 (c+d x)^{3/4}}{3 (b c-a d) (a+b x)^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=-\frac {4 (c+d x)^{3/4}}{3 (b c-a d) (a+b x)^{3/4}} \]

[In]

Integrate[1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x]

[Out]

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

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {4 \left (d x +c \right )^{\frac {3}{4}}}{3 \left (b x +a \right )^{\frac {3}{4}} \left (a d -b c \right )}\)	\(27\)

[In]

int(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=-\frac {4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{3 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )}} \]

[In]

integrate(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {7}{4}} \sqrt [4]{c + d x}}\, dx \]

[In]

integrate(1/(b*x+a)**(7/4)/(d*x+c)**(1/4),x)

[Out]

Integral(1/((a + b*x)**(7/4)*(c + d*x)**(1/4)), x)

Maxima [F]

\[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/4)*(d*x + c)^(1/4)), x)

Giac [F]

\[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/4)*(d*x + c)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{7/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]

[In]

int(1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x)

[Out]

int(1/((a + b*x)^(7/4)*(c + d*x)^(1/4)), x)