3.12.80 \(\int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx\) [1180]

Optimal. Leaf size=33 \[ -\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]

[Out]

-2/3*I*(a+I*a*x)^(3/4)/a^2/(a-I*a*x)^(3/4)

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Rubi [A]
time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {37} \begin {gather*} -\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-2*I)/3)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*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*} \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 33, normalized size = 1.00 \begin {gather*} -\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-2*I)/3)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(3/4))

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Maple [A]
time = 0.16, size = 31, normalized size = 0.94

method	result	size
risch	\(\frac {\frac {2 x}{3}-\frac {2 i}{3}}{a \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

2/3/a/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(x-I)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)), x)

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Fricas [A]
time = 0.86, size = 31, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{3 \, {\left (a^{3} x + i \, a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")

[Out]

2/3*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)/(a^3*x + I*a^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(1/4),x)

[Out]

Integral(1/((I*a*(x - I))**(1/4)*(-I*a*(x + I))**(7/4)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:ext_reduce Error: Bad Argument TypeDone

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Mupad [B]
time = 0.55, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2\,\left (x-\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{3\,a^2\,\left (-1+x\,1{}\mathrm {i}\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - a*x*1i)^(7/4)*(a + a*x*1i)^(1/4)),x)

[Out]

-(2*(x - 1i)*(-a*(x*1i - 1))^(1/4))/(3*a^2*(x*1i - 1)*(a*(x*1i + 1))^(1/4))

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