3.12.12 \(\int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=100 \[ -\frac {16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \]

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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}}{45 a^4 \sqrt [4]{a-i a x}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

(((-2*I)/9)*(a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(9/4)) - (((8*I)/45)*(a + I*a*x)^(1/4))/(a^3*(a - I*a*x)^(5/4)
) - (((16*I)/45)*(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)^{13/4} (a+i a x)^{3/4}} \, dx &=-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}+\frac {4 \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{3/4}} \, dx}{9 a}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx}{45 a^2}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac {16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 52, normalized size = 0.52 \begin {gather*} \frac {2 \left (-8 i x^2+20 x+17 i\right ) \sqrt [4]{a+i a x}}{45 a^4 (x+i)^2 \sqrt [4]{a-i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

(2*(a + I*a*x)^(1/4)*(17*I + 20*x - (8*I)*x^2))/(45*a^4*(I + x)^2*(a - I*a*x)^(1/4))

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IntegrateAlgebraic [A]  time = 0.12, size = 77, normalized size = 0.77 \begin {gather*} -\frac {i \sqrt [4]{a+i a x} \left (\frac {5 (a+i a x)^2}{(a-i a x)^2}+\frac {18 (a+i a x)}{a-i a x}+45\right )}{90 a^4 \sqrt [4]{a-i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

((-1/90*I)*(a + I*a*x)^(1/4)*(45 + (18*(a + I*a*x))/(a - I*a*x) + (5*(a + I*a*x)^2)/(a - I*a*x)^2))/(a^4*(a -
I*a*x)^(1/4))

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fricas [A]  time = 1.40, size = 58, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (8 \, x^{2} + 20 i \, x - 17\right )}}{45 \, a^{5} x^{3} + 135 i \, a^{5} x^{2} - 135 \, a^{5} x - 45 i \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(3/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4)*(8*x^2 + 20*I*x - 17)/(45*a^5*x^3 + 135*I*a^5*x^2 - 135*a^5*x - 45*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 {3}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(13/4)), x)

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maple [A]  time = 0.06, size = 50, normalized size = 0.50 \begin {gather*} \frac {\frac {16}{45} x^{3}+\frac {8}{15} i x^{2}+\frac {2}{15} x +\frac {34}{45} i}{\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 )^{2} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-I*a*x+a)^(13/4)/(I*a*x+a)^(3/4),x)

[Out]

2/45/a^3/((I*x+1)*a)^(3/4)/(-(I*x-1)*a)^(1/4)*(12*I*x^2+8*x^3+3*x+17*I)/(x+I)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(13/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(3/4)),x)

[Out]

int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(3/4)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(3/4),x)

[Out]

Timed out

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