3.12.7 \(\int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=133 \[ -\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \]

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Rubi [A]  time = 0.03, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-2*I)/15)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(15/4)) - (((4*I)/55)*(a + I*a*x)^(3/4))/(a^3*(a - I*a*x)^(11
/4)) - (((16*I)/385)*(a + I*a*x)^(3/4))/(a^4*(a - I*a*x)^(7/4)) - (((32*I)/1155)*(a + I*a*x)^(3/4))/(a^5*(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]

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)^{19/4} \sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}+\frac {2 \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx}{5 a}\\ &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}+\frac {8 \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx}{55 a^2}\\ &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}+\frac {16 \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{385 a^3}\\ &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 57, normalized size = 0.43 \begin {gather*} \frac {2 \left (-16 i x^3+72 x^2+138 i x-159\right ) (a+i a x)^{3/4}}{1155 a^5 (x+i)^3 (a-i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

(2*(a + I*a*x)^(3/4)*(-159 + (138*I)*x + 72*x^2 - (16*I)*x^3))/(1155*a^5*(I + x)^3*(a - I*a*x)^(3/4))

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IntegrateAlgebraic [A]  time = 0.12, size = 99, normalized size = 0.74 \begin {gather*} -\frac {i (a+i a x)^{15/4} \left (\frac {385 (a-i a x)^3}{(a+i a x)^3}+\frac {495 (a-i a x)^2}{(a+i a x)^2}+\frac {315 (a-i a x)}{a+i a x}+77\right )}{4620 a^5 (a-i a x)^{15/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

((-1/4620*I)*(a + I*a*x)^(15/4)*(77 + (385*(a - I*a*x)^3)/(a + I*a*x)^3 + (495*(a - I*a*x)^2)/(a + I*a*x)^2 +
(315*(a - I*a*x))/(a + I*a*x)))/(a^5*(a - I*a*x)^(15/4))

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fricas [A]  time = 1.47, size = 70, normalized size = 0.53 \begin {gather*} \frac {{\left (32 \, x^{3} + 144 i \, x^{2} - 276 \, x - 318 i\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{1155 \, a^{6} x^{4} + 4620 i \, a^{6} x^{3} - 6930 \, a^{6} x^{2} - 4620 i \, a^{6} x + 1155 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")

[Out]

(32*x^3 + 144*I*x^2 - 276*x - 318*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)/(1155*a^6*x^4 + 4620*I*a^6*x^3 - 693
0*a^6*x^2 - 4620*I*a^6*x + 1155*a^6)

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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 {1}{4}} {\left (-i \, a x + a\right )}^{\frac {19}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)), x)

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maple [A]  time = 0.06, size = 55, normalized size = 0.41 \begin {gather*} \frac {\frac {32}{1155} x^{4}+\frac {16}{165} i x^{3}-\frac {4}{35} x^{2}-\frac {2}{55} i x -\frac {106}{385}}{\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 )^{3} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-I*a*x+a)^(19/4)/(I*a*x+a)^(1/4),x)

[Out]

2/1155/a^4/(-(I*x-1)*a)^(3/4)/((I*x+1)*a)^(1/4)*(56*I*x^3+16*x^4-21*I*x-159-66*x^2)/(x+I)^3

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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 {1}{4}} {\left (-i \, a x + a\right )}^{\frac {19}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)), x)

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mupad [B]  time = 0.79, size = 57, normalized size = 0.43 \begin {gather*} -\frac {{\left (x-\mathrm {i}\right )}^5\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (-16\,x^3-x^2\,72{}\mathrm {i}+138\,x+159{}\mathrm {i}\right )\,2{}\mathrm {i}}{1155\,a^5\,{\left (x^2+1\right )}^4\,{\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)^(19/4)*(a + a*x*1i)^(1/4)),x)

[Out]

-((x - 1i)^5*(-a*(x*1i - 1))^(1/4)*(138*x - x^2*72i - 16*x^3 + 159i)*2i)/(1155*a^5*(x^2 + 1)^4*(a*(x*1i + 1))^
(1/4))

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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)**(19/4)/(a+I*a*x)**(1/4),x)

[Out]

Timed out

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