3.15.88 \(\int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx\)

Optimal. Leaf size=101 \[ -\frac {128 d^2 (c+d x)^{9/4}}{1989 (a+b x)^{9/4} (b c-a d)^3}+\frac {32 d (c+d x)^{9/4}}{221 (a+b x)^{13/4} (b c-a d)^2}-\frac {4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)} \]

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Rubi [A]  time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {128 d^2 (c+d x)^{9/4}}{1989 (a+b x)^{9/4} (b c-a d)^3}+\frac {32 d (c+d x)^{9/4}}{221 (a+b x)^{13/4} (b c-a d)^2}-\frac {4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/4)/(a + b*x)^(21/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(17*(b*c - a*d)*(a + b*x)^(17/4)) + (32*d*(c + d*x)^(9/4))/(221*(b*c - a*d)^2*(a + b*x)^(
13/4)) - (128*d^2*(c + d*x)^(9/4))/(1989*(b*c - a*d)^3*(a + b*x)^(9/4))

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 {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx &=-\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}-\frac {(8 d) \int \frac {(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx}{17 (b c-a d)}\\ &=-\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac {32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}+\frac {\left (32 d^2\right ) \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx}{221 (b c-a d)^2}\\ &=-\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac {32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}-\frac {128 d^2 (c+d x)^{9/4}}{1989 (b c-a d)^3 (a+b x)^{9/4}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 77, normalized size = 0.76 \begin {gather*} -\frac {4 (c+d x)^{9/4} \left (221 a^2 d^2+34 a b d (4 d x-9 c)+b^2 \left (117 c^2-72 c d x+32 d^2 x^2\right )\right )}{1989 (a+b x)^{17/4} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/4)/(a + b*x)^(21/4),x]

[Out]

(-4*(c + d*x)^(9/4)*(221*a^2*d^2 + 34*a*b*d*(-9*c + 4*d*x) + b^2*(117*c^2 - 72*c*d*x + 32*d^2*x^2)))/(1989*(b*
c - a*d)^3*(a + b*x)^(17/4))

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IntegrateAlgebraic [A]  time = 0.18, size = 73, normalized size = 0.72 \begin {gather*} -\frac {4 (c+d x)^{9/4} \left (\frac {117 b^2 (c+d x)^2}{(a+b x)^2}-\frac {306 b d (c+d x)}{a+b x}+221 d^2\right )}{1989 (a+b x)^{9/4} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c + d*x)^(5/4)/(a + b*x)^(21/4),x]

[Out]

(-4*(c + d*x)^(9/4)*(221*d^2 - (306*b*d*(c + d*x))/(a + b*x) + (117*b^2*(c + d*x)^2)/(a + b*x)^2))/(1989*(b*c
- a*d)^3*(a + b*x)^(9/4))

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fricas [B]  time = 1.52, size = 426, normalized size = 4.22 \begin {gather*} -\frac {4 \, {\left (32 \, b^{2} d^{4} x^{4} + 117 \, b^{2} c^{4} - 306 \, a b c^{3} d + 221 \, a^{2} c^{2} d^{2} - 8 \, {\left (b^{2} c d^{3} - 17 \, a b d^{4}\right )} x^{3} + {\left (5 \, b^{2} c^{2} d^{2} - 34 \, a b c d^{3} + 221 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (81 \, b^{2} c^{3} d - 238 \, a b c^{2} d^{2} + 221 \, a^{2} c d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{1989 \, {\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(21/4),x, algorithm="fricas")

[Out]

-4/1989*(32*b^2*d^4*x^4 + 117*b^2*c^4 - 306*a*b*c^3*d + 221*a^2*c^2*d^2 - 8*(b^2*c*d^3 - 17*a*b*d^4)*x^3 + (5*
b^2*c^2*d^2 - 34*a*b*c*d^3 + 221*a^2*d^4)*x^2 + 2*(81*b^2*c^3*d - 238*a*b*c^2*d^2 + 221*a^2*c*d^3)*x)*(b*x + a
)^(3/4)*(d*x + c)^(1/4)/(a^5*b^3*c^3 - 3*a^6*b^2*c^2*d + 3*a^7*b*c*d^2 - a^8*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d +
3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^5 + 5*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^4 + 10*
(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^3 + 10*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^
5*b^3*c*d^2 - a^6*b^2*d^3)*x^2 + 5*(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {21}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(21/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(21/4), x)

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maple [A]  time = 0.01, size = 105, normalized size = 1.04 \begin {gather*} \frac {4 \left (d x +c \right )^{\frac {9}{4}} \left (32 b^{2} x^{2} d^{2}+136 a b \,d^{2} x -72 b^{2} c d x +221 a^{2} d^{2}-306 a b c d +117 b^{2} c^{2}\right )}{1989 \left (b x +a \right )^{\frac {17}{4}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/4)/(b*x+a)^(21/4),x)

[Out]

4/1989*(d*x+c)^(9/4)*(32*b^2*d^2*x^2+136*a*b*d^2*x-72*b^2*c*d*x+221*a^2*d^2-306*a*b*c*d+117*b^2*c^2)/(b*x+a)^(
17/4)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {21}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(21/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(21/4), x)

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mupad [B]  time = 1.13, size = 268, normalized size = 2.65 \begin {gather*} \frac {{\left (c+d\,x\right )}^{1/4}\,\left (\frac {884\,a^2\,c^2\,d^2-1224\,a\,b\,c^3\,d+468\,b^2\,c^4}{1989\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,\left (884\,a^2\,d^4-136\,a\,b\,c\,d^3+20\,b^2\,c^2\,d^2\right )}{1989\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {128\,d^4\,x^4}{1989\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {32\,d^3\,x^3\,\left (17\,a\,d-b\,c\right )}{1989\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,c\,d\,x\,\left (221\,a^2\,d^2-238\,a\,b\,c\,d+81\,b^2\,c^2\right )}{1989\,b^4\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4\,{\left (a+b\,x\right )}^{1/4}+\frac {a^4\,{\left (a+b\,x\right )}^{1/4}}{b^4}+\frac {6\,a^2\,x^2\,{\left (a+b\,x\right )}^{1/4}}{b^2}+\frac {4\,a\,x^3\,{\left (a+b\,x\right )}^{1/4}}{b}+\frac {4\,a^3\,x\,{\left (a+b\,x\right )}^{1/4}}{b^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/4)/(a + b*x)^(21/4),x)

[Out]

((c + d*x)^(1/4)*((468*b^2*c^4 + 884*a^2*c^2*d^2 - 1224*a*b*c^3*d)/(1989*b^4*(a*d - b*c)^3) + (x^2*(884*a^2*d^
4 + 20*b^2*c^2*d^2 - 136*a*b*c*d^3))/(1989*b^4*(a*d - b*c)^3) + (128*d^4*x^4)/(1989*b^2*(a*d - b*c)^3) + (32*d
^3*x^3*(17*a*d - b*c))/(1989*b^3*(a*d - b*c)^3) + (8*c*d*x*(221*a^2*d^2 + 81*b^2*c^2 - 238*a*b*c*d))/(1989*b^4
*(a*d - b*c)^3)))/(x^4*(a + b*x)^(1/4) + (a^4*(a + b*x)^(1/4))/b^4 + (6*a^2*x^2*(a + b*x)^(1/4))/b^2 + (4*a*x^
3*(a + b*x)^(1/4))/b + (4*a^3*x*(a + b*x)^(1/4))/b^3)

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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((d*x+c)**(5/4)/(b*x+a)**(21/4),x)

[Out]

Timed out

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