3.16.1 \(\int \frac {1}{(a+b x)^{9/4} (c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=66 \[ \frac {16 d \sqrt [4]{c+d x}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac {4 \sqrt [4]{c+d x}}{5 (a+b x)^{5/4} (b c-a d)} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {16 d \sqrt [4]{c+d x}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac {4 \sqrt [4]{c+d x}}{5 (a+b x)^{5/4} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(9/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*(c + d*x)^(1/4))/(5*(b*c - a*d)*(a + b*x)^(5/4)) + (16*d*(c + d*x)^(1/4))/(5*(b*c - a*d)^2*(a + b*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+b x)^{9/4} (c+d x)^{3/4}} \, dx &=-\frac {4 \sqrt [4]{c+d x}}{5 (b c-a d) (a+b x)^{5/4}}-\frac {(4 d) \int \frac {1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx}{5 (b c-a d)}\\ &=-\frac {4 \sqrt [4]{c+d x}}{5 (b c-a d) (a+b x)^{5/4}}+\frac {16 d \sqrt [4]{c+d x}}{5 (b c-a d)^2 \sqrt [4]{a+b x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {4 \sqrt [4]{c+d x} (5 a d-b c+4 b d x)}{5 (a+b x)^{5/4} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(9/4)*(c + d*x)^(3/4)),x]

[Out]

(4*(c + d*x)^(1/4)*(-(b*c) + 5*a*d + 4*b*d*x))/(5*(b*c - a*d)^2*(a + b*x)^(5/4))

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IntegrateAlgebraic [A]  time = 0.12, size = 56, normalized size = 0.85 \begin {gather*} -\frac {4 \left (\frac {b (c+d x)^{5/4}}{(a+b x)^{5/4}}-\frac {5 d \sqrt [4]{c+d x}}{\sqrt [4]{a+b x}}\right )}{5 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(9/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*((-5*d*(c + d*x)^(1/4))/(a + b*x)^(1/4) + (b*(c + d*x)^(5/4))/(a + b*x)^(5/4)))/(5*(b*c - a*d)^2)

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fricas [B]  time = 1.21, size = 118, normalized size = 1.79 \begin {gather*} \frac {4 \, {\left (4 \, b d x - b c + 5 \, a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{5 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/4)/(d*x+c)^(3/4),x, algorithm="fricas")

[Out]

4/5*(4*b*d*x - b*c + 5*a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 -
2*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/4)/(d*x+c)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(9/4)*(d*x + c)^(3/4)), x)

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maple [A]  time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} \frac {4 \left (d x +c \right )^{\frac {1}{4}} \left (4 b d x +5 a d -b c \right )}{5 \left (b x +a \right )^{\frac {5}{4}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(9/4)/(d*x+c)^(3/4),x)

[Out]

4/5*(d*x+c)^(1/4)*(4*b*d*x+5*a*d-b*c)/(b*x+a)^(5/4)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/4)/(d*x+c)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(9/4)*(d*x + c)^(3/4)), x)

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mupad [B]  time = 0.87, size = 71, normalized size = 1.08 \begin {gather*} \frac {\left (\frac {16\,d\,x}{5\,{\left (a\,d-b\,c\right )}^2}+\frac {20\,a\,d-4\,b\,c}{5\,b\,{\left (a\,d-b\,c\right )}^2}\right )\,{\left (c+d\,x\right )}^{1/4}}{x\,{\left (a+b\,x\right )}^{1/4}+\frac {a\,{\left (a+b\,x\right )}^{1/4}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(9/4)*(c + d*x)^(3/4)),x)

[Out]

(((16*d*x)/(5*(a*d - b*c)^2) + (20*a*d - 4*b*c)/(5*b*(a*d - b*c)^2))*(c + d*x)^(1/4))/(x*(a + b*x)^(1/4) + (a*
(a + b*x)^(1/4))/b)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {9}{4}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(9/4)/(d*x+c)**(3/4),x)

[Out]

Integral(1/((a + b*x)**(9/4)*(c + d*x)**(3/4)), x)

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