3.15.96 \(\int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=136 \[ \frac {512 d^3 (c+d x)^{3/4}}{1155 (a+b x)^{3/4} (b c-a d)^4}-\frac {128 d^2 (c+d x)^{3/4}}{385 (a+b x)^{7/4} (b c-a d)^3}+\frac {16 d (c+d x)^{3/4}}{55 (a+b x)^{11/4} (b c-a d)^2}-\frac {4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)} \]

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Rubi [A]  time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {512 d^3 (c+d x)^{3/4}}{1155 (a+b x)^{3/4} (b c-a d)^4}-\frac {128 d^2 (c+d x)^{3/4}}{385 (a+b x)^{7/4} (b c-a d)^3}+\frac {16 d (c+d x)^{3/4}}{55 (a+b x)^{11/4} (b c-a d)^2}-\frac {4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x]

[Out]

(-4*(c + d*x)^(3/4))/(15*(b*c - a*d)*(a + b*x)^(15/4)) + (16*d*(c + d*x)^(3/4))/(55*(b*c - a*d)^2*(a + b*x)^(1
1/4)) - (128*d^2*(c + d*x)^(3/4))/(385*(b*c - a*d)^3*(a + b*x)^(7/4)) + (512*d^3*(c + d*x)^(3/4))/(1155*(b*c -
a*d)^4*(a + b*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+b x)^{19/4} \sqrt [4]{c+d x}} \, dx &=-\frac {4 (c+d x)^{3/4}}{15 (b c-a d) (a+b x)^{15/4}}-\frac {(4 d) \int \frac {1}{(a+b x)^{15/4} \sqrt [4]{c+d x}} \, dx}{5 (b c-a d)}\\ &=-\frac {4 (c+d x)^{3/4}}{15 (b c-a d) (a+b x)^{15/4}}+\frac {16 d (c+d x)^{3/4}}{55 (b c-a d)^2 (a+b x)^{11/4}}+\frac {\left (32 d^2\right ) \int \frac {1}{(a+b x)^{11/4} \sqrt [4]{c+d x}} \, dx}{55 (b c-a d)^2}\\ &=-\frac {4 (c+d x)^{3/4}}{15 (b c-a d) (a+b x)^{15/4}}+\frac {16 d (c+d x)^{3/4}}{55 (b c-a d)^2 (a+b x)^{11/4}}-\frac {128 d^2 (c+d x)^{3/4}}{385 (b c-a d)^3 (a+b x)^{7/4}}-\frac {\left (128 d^3\right ) \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx}{385 (b c-a d)^3}\\ &=-\frac {4 (c+d x)^{3/4}}{15 (b c-a d) (a+b x)^{15/4}}+\frac {16 d (c+d x)^{3/4}}{55 (b c-a d)^2 (a+b x)^{11/4}}-\frac {128 d^2 (c+d x)^{3/4}}{385 (b c-a d)^3 (a+b x)^{7/4}}+\frac {512 d^3 (c+d x)^{3/4}}{1155 (b c-a d)^4 (a+b x)^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 118, normalized size = 0.87 \begin {gather*} \frac {4 (c+d x)^{3/4} \left (385 a^3 d^3+165 a^2 b d^2 (4 d x-3 c)+15 a b^2 d \left (21 c^2-24 c d x+32 d^2 x^2\right )+b^3 \left (-77 c^3+84 c^2 d x-96 c d^2 x^2+128 d^3 x^3\right )\right )}{1155 (a+b x)^{15/4} (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x]

[Out]

(4*(c + d*x)^(3/4)*(385*a^3*d^3 + 165*a^2*b*d^2*(-3*c + 4*d*x) + 15*a*b^2*d*(21*c^2 - 24*c*d*x + 32*d^2*x^2) +
b^3*(-77*c^3 + 84*c^2*d*x - 96*c*d^2*x^2 + 128*d^3*x^3)))/(1155*(b*c - a*d)^4*(a + b*x)^(15/4))

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IntegrateAlgebraic [A]  time = 0.18, size = 95, normalized size = 0.70 \begin {gather*} \frac {4 (c+d x)^{15/4} \left (\frac {315 b^2 d (a+b x)}{c+d x}+\frac {385 d^3 (a+b x)^3}{(c+d x)^3}-\frac {495 b d^2 (a+b x)^2}{(c+d x)^2}-77 b^3\right )}{1155 (a+b x)^{15/4} (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x]

[Out]

(4*(c + d*x)^(15/4)*(-77*b^3 + (385*d^3*(a + b*x)^3)/(c + d*x)^3 - (495*b*d^2*(a + b*x)^2)/(c + d*x)^2 + (315*
b^2*d*(a + b*x))/(c + d*x)))/(1155*(b*c - a*d)^4*(a + b*x)^(15/4))

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fricas [B]  time = 5.10, size = 419, normalized size = 3.08 \begin {gather*} \frac {4 \, {\left (128 \, b^{3} d^{3} x^{3} - 77 \, b^{3} c^{3} + 315 \, a b^{2} c^{2} d - 495 \, a^{2} b c d^{2} + 385 \, a^{3} d^{3} - 96 \, {\left (b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} + 12 \, {\left (7 \, b^{3} c^{2} d - 30 \, a b^{2} c d^{2} + 55 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{1155 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

4/1155*(128*b^3*d^3*x^3 - 77*b^3*c^3 + 315*a*b^2*c^2*d - 495*a^2*b*c*d^2 + 385*a^3*d^3 - 96*(b^3*c*d^2 - 5*a*b
^2*d^3)*x^2 + 12*(7*b^3*c^2*d - 30*a*b^2*c*d^2 + 55*a^2*b*d^3)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4)/(a^4*b^4*c^4
- 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^
2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^4 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3
+ a^5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2
+ 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4)*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 {19}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(19/4)*(d*x + c)^(1/4)), x)

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maple [A]  time = 0.01, size = 171, normalized size = 1.26 \begin {gather*} \frac {4 \left (d x +c \right )^{\frac {3}{4}} \left (128 b^{3} d^{3} x^{3}+480 a \,b^{2} d^{3} x^{2}-96 b^{3} c \,d^{2} x^{2}+660 a^{2} b \,d^{3} x -360 a \,b^{2} c \,d^{2} x +84 b^{3} c^{2} d x +385 a^{3} d^{3}-495 a^{2} b c \,d^{2}+315 a \,b^{2} c^{2} d -77 b^{3} c^{3}\right )}{1155 \left (b x +a \right )^{\frac {15}{4}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x)

[Out]

4/1155*(d*x+c)^(3/4)*(128*b^3*d^3*x^3+480*a*b^2*d^3*x^2-96*b^3*c*d^2*x^2+660*a^2*b*d^3*x-360*a*b^2*c*d^2*x+84*
b^3*c^2*d*x+385*a^3*d^3-495*a^2*b*c*d^2+315*a*b^2*c^2*d-77*b^3*c^3)/(b*x+a)^(15/4)/(a^4*d^4-4*a^3*b*c*d^3+6*a^
2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(19/4)*(d*x + c)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{19/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x)

[Out]

int(1/((a + b*x)^(19/4)*(c + d*x)^(1/4)), x)

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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/(b*x+a)**(19/4)/(d*x+c)**(1/4),x)

[Out]

Timed out

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