3.1683 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx\)

Optimal. Leaf size=66 \[ \frac{16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(9/4))/(13*(b*c - a*d)*(a + b*x)^(13/4)) + (16*d*(c + d*x)^(9/4))/(117*(b*c - a*d)^2*(a + b*x)^(
9/4))

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Rubi [A]  time = 0.0093824, antiderivative size = 66, normalized size of antiderivative = 1., 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} \[ \frac{16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/4)/(a + b*x)^(17/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(13*(b*c - a*d)*(a + b*x)^(13/4)) + (16*d*(c + d*x)^(9/4))/(117*(b*c - a*d)^2*(a + b*x)^(
9/4))

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])

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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx &=-\frac{4 (c+d x)^{9/4}}{13 (b c-a d) (a+b x)^{13/4}}-\frac{(4 d) \int \frac{(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx}{13 (b c-a d)}\\ &=-\frac{4 (c+d x)^{9/4}}{13 (b c-a d) (a+b x)^{13/4}}+\frac{16 d (c+d x)^{9/4}}{117 (b c-a d)^2 (a+b x)^{9/4}}\\ \end{align*}

Mathematica [A]  time = 0.0259256, size = 46, normalized size = 0.7 \[ \frac{4 (c+d x)^{9/4} (13 a d-9 b c+4 b d x)}{117 (a+b x)^{13/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/4)/(a + b*x)^(17/4),x]

[Out]

(4*(c + d*x)^(9/4)*(-9*b*c + 13*a*d + 4*b*d*x))/(117*(b*c - a*d)^2*(a + b*x)^(13/4))

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Maple [A]  time = 0.005, size = 54, normalized size = 0.8 \begin{align*}{\frac{16\,bdx+52\,ad-36\,bc}{117\,{a}^{2}{d}^{2}-234\,abcd+117\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{9}{4}}} \left ( bx+a \right ) ^{-{\frac{13}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/4)/(b*x+a)^(17/4),x)

[Out]

4/117*(d*x+c)^(9/4)*(4*b*d*x+13*a*d-9*b*c)/(b*x+a)^(13/4)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{17}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(17/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(17/4), x)

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Fricas [B]  time = 2.98466, size = 494, normalized size = 7.48 \begin{align*} \frac{4 \,{\left (4 \, b d^{3} x^{3} - 9 \, b c^{3} + 13 \, a c^{2} d -{\left (b c d^{2} - 13 \, a d^{3}\right )} x^{2} - 2 \,{\left (7 \, b c^{2} d - 13 \, a c d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{117 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(17/4),x, algorithm="fricas")

[Out]

4/117*(4*b*d^3*x^3 - 9*b*c^3 + 13*a*c^2*d - (b*c*d^2 - 13*a*d^3)*x^2 - 2*(7*b*c^2*d - 13*a*c*d^2)*x)*(b*x + a)
^(3/4)*(d*x + c)^(1/4)/(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^4 + 4*(a
*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^3 + 6*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x^2 + 4*(a^3*b^3*c
^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/4)/(b*x+a)**(17/4),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{17}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(17/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(17/4), x)