3.12.22 \(\int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=100 \[ \frac {16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 \sqrt [4]{a+i a x} (a-i a x)^{3/4}}-\frac {2 i}{7 a^2 \sqrt [4]{a+i a x} (a-i a x)^{7/4}} \]

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Rubi [A]  time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {45, 37} \begin {gather*} \frac {16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 \sqrt [4]{a+i a x} (a-i a x)^{3/4}}-\frac {2 i}{7 a^2 \sqrt [4]{a+i a x} (a-i a x)^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(5/4)),x]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)) - ((8*I)/21)/(a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)) +
(((16*I)/21)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(1/4))

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]

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{5/4}} \, dx &=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}+\frac {4 \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx}{7 a}\\ &=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{21 a^2}\\ &=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 0.50 \begin {gather*} \frac {16 x^2+24 i x-2}{21 a^3 (x+i) (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(5/4)),x]

[Out]

(-2 + (24*I)*x + 16*x^2)/(21*a^3*(I + x)*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4))

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IntegrateAlgebraic [A]  time = 0.14, size = 77, normalized size = 0.77 \begin {gather*} \frac {i (a+i a x)^{7/4} \left (\frac {21 (a-i a x)^2}{(a+i a x)^2}-\frac {14 (a-i a x)}{a+i a x}-3\right )}{42 a^4 (a-i a x)^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(5/4)),x]

[Out]

((I/42)*(a + I*a*x)^(7/4)*(-3 + (21*(a - I*a*x)^2)/(a + I*a*x)^2 - (14*(a - I*a*x))/(a + I*a*x)))/(a^4*(a - I*
a*x)^(7/4))

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fricas [A]  time = 0.88, size = 58, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} + 12 i \, x - 1\right )}}{21 \, a^{5} x^{3} + 21 i \, a^{5} x^{2} + 21 \, a^{5} x + 21 i \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(5/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(8*x^2 + 12*I*x - 1)/(21*a^5*x^3 + 21*I*a^5*x^2 + 21*a^5*x + 21*I*a^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {5}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(11/4)), x)

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maple [A]  time = 0.06, size = 44, normalized size = 0.44 \begin {gather*} \frac {\frac {16}{21} x^{2}+\frac {8}{7} i x -\frac {2}{21}}{\left (-\left (i x -1\right ) a \right )^{\frac {3}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} \left (x +i\right ) a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-I*a*x+a)^(11/4)/(I*a*x+a)^(5/4),x)

[Out]

2/21/a^3/(-(I*x-1)*a)^(3/4)/((I*x+1)*a)^(1/4)*(8*x^2+12*I*x-1)/(x+I)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(5/4),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 0.76, size = 46, normalized size = 0.46 \begin {gather*} -\frac {{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (8\,x^2+x\,12{}\mathrm {i}-1\right )\,2{}\mathrm {i}}{21\,a^4\,{\left (-1+x\,1{}\mathrm {i}\right )}^2\,{\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)^(11/4)*(a + a*x*1i)^(5/4)),x)

[Out]

-((-a*(x*1i - 1))^(1/4)*(x*12i + 8*x^2 - 1)*2i)/(21*a^4*(x*1i - 1)^2*(a*(x*1i + 1))^(1/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}} \left (- i a \left (x + i\right )\right )^{\frac {11}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(5/4),x)

[Out]

Integral(1/((I*a*(x - I))**(5/4)*(-I*a*(x + I))**(11/4)), x)

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