∫ 1_______ (a+bx)11∕4(c+dx)5∕4 dx [1721]

Optimal. Leaf size=101 \[ -\frac {4}{7 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}+\frac {32 d}{21 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}+\frac {128 d^2 \sqrt [4]{a+b x}}{21 (b c-a d)^3 \sqrt [4]{c+d x}} \]

-4/7/(-a*d+b*c)/(b*x+a)^(7/4)/(d*x+c)^(1/4)+32/21*d/(-a*d+b*c)^2/(b*x+a)^(3/4)/(d*x+c)^(1/4)+128/21*d^2*(b*x+a

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Rubi [A]
time = 0.01, 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 = {47, 37} \begin {gather*} \frac {128 d^2 \sqrt [4]{a+b x}}{21 \sqrt [4]{c+d x} (b c-a d)^3}+\frac {32 d}{21 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}-\frac {4}{7 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)} \end {gather*}

-4/(7*(b*c - a*d)*(a + b*x)^(7/4)*(c + d*x)^(1/4)) + (32*d)/(21*(b*c - a*d)^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))

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

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]))) && (

\begin {align*} \int \frac {1}{(a+b x)^{11/4} (c+d x)^{5/4}} \, dx &=-\frac {4}{7 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}-\frac {(8 d) \int \frac {1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx}{7 (b c-a d)}\\ &=-\frac {4}{7 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}+\frac {32 d}{21 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}+\frac {\left (32 d^2\right ) \int \frac {1}{(a+b x)^{3/4} (c+d x)^{5/4}} \, dx}{21 (b c-a d)^2}\\ &=-\frac {4}{7 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}+\frac {32 d}{21 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}+\frac {128 d^2 \sqrt [4]{a+b x}}{21 (b c-a d)^3 \sqrt [4]{c+d x}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 73, normalized size = 0.72 \begin {gather*} -\frac {4 (c+d x)^{7/4} \left (3 b^2-\frac {21 d^2 (a+b x)^2}{(c+d x)^2}-\frac {14 b d (a+b x)}{c+d x}\right )}{21 (b c-a d)^3 (a+b x)^{7/4}} \end {gather*}

(-4*(c + d*x)^(7/4)*(3*b^2 - (21*d^2*(a + b*x)^2)/(c + d*x)^2 - (14*b*d*(a + b*x))/(c + d*x)))/(21*(b*c - a*d)

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method	result	size

gosper	\(-\frac {4 \left (32 b^{2} x^{2} d^{2}+56 a b \,d^{2} x +8 b^{2} c d x +21 a^{2} d^{2}+14 a b c d -3 b^{2} c^{2}\right )}{21 \left (b x +a \right )^{\frac {7}{4}} \left (d x +c \right )^{\frac {1}{4}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(105\)

-4/21*(32*b^2*d^2*x^2+56*a*b*d^2*x+8*b^2*c*d*x+21*a^2*d^2+14*a*b*c*d-3*b^2*c^2)/(b*x+a)^(7/4)/(d*x+c)^(1/4)/(a

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

integrate(1/(b*x+a)^(11/4)/(d*x+c)^(5/4),x, algorithm="maxima")

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (83) = 166\).
time = 0.73, size = 273, normalized size = 2.70 \begin {gather*} \frac {4 \, {\left (32 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 14 \, a b c d + 21 \, a^{2} d^{2} + 8 \, {\left (b^{2} c d + 7 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{21 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \end {gather*}

integrate(1/(b*x+a)^(11/4)/(d*x+c)^(5/4),x, algorithm="fricas")

4/21*(32*b^2*d^2*x^2 - 3*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2 + 8*(b^2*c*d + 7*a*b*d^2)*x)*(b*x + a)^(1/4)*(d*x +

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

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

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

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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 {11}{4}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \end {gather*}

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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 )}^{11/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \end {gather*}

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3.18.21 \(\int \frac {1}{(a+b x)^{11/4} (c+d x)^{5/4}} \, dx\) [1721]