3.12.26 \(\int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=100 \[ \frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac {2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]

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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}}{15 a^4 \sqrt [4]{a+i a x}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac {2 i}{3 a^2 (a+i a x)^{5/4} (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)^(9/4)),x]

[Out]

((-2*I)/3)/(a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(5/4)) + (((8*I)/15)*(a - I*a*x)^(1/4))/(a^3*(a + I*a*x)^(5/4))
+ (((16*I)/15)*(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)^{7/4} (a+i a x)^{9/4}} \, dx &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {4 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{3 a}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{15 a^2}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{15 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 {2 \left (8 x^2-4 i x+7\right )}{15 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)^(7/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*(7 - (4*I)*x + 8*x^2))/(15*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)^{3/4} \left (\frac {3 (a-i a x)^2}{(a+i a x)^2}+\frac {30 (a-i a x)}{a+i a x}-5\right )}{30 a^4 (a-i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(9/4)),x]

[Out]

((I/30)*(a + I*a*x)^(3/4)*(-5 + (3*(a - I*a*x)^2)/(a + I*a*x)^2 + (30*(a - I*a*x))/(a + I*a*x)))/(a^4*(a - I*a
*x)^(3/4))

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fricas [A]  time = 1.27, 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} - 4 i \, x + 7\right )}}{15 \, a^{5} x^{3} - 15 i \, a^{5} x^{2} + 15 \, a^{5} x - 15 i \, a^{5}} \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)^(9/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(8*x^2 - 4*I*x + 7)/(15*a^5*x^3 - 15*I*a^5*x^2 + 15*a^5*x - 15*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 {9}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \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)^(9/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(7/4)), x)

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maple [A]  time = 0.06, size = 44, normalized size = 0.44 \begin {gather*} \frac {\frac {16}{15} x^{2}-\frac {8}{15} i x +\frac {14}{15}}{\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)^(7/4)/(I*a*x+a)^(9/4),x)

[Out]

2/15/a^3/(-(I*x-1)*a)^(3/4)/((I*x+1)*a)^(1/4)*(8*x^2-4*I*x+7)/(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)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.53, size = 45, normalized size = 0.45 \begin {gather*} \frac {2\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (x^2\,8{}\mathrm {i}+4\,x+7{}\mathrm {i}\right )}{15\,a^4\,\left (x^2+1\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)^(9/4)),x)

[Out]

(2*(-a*(x*1i - 1))^(1/4)*(4*x + x^2*8i + 7i))/(15*a^4*(x^2 + 1)*(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 {9}{4}} \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)**(9/4),x)

[Out]

Integral(1/((I*a*(x - I))**(9/4)*(-I*a*(x + I))**(7/4)), x)

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