Optimal. Leaf size=66 \[ -\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}+\frac {9 d \sqrt [3]{c+d x}}{4 (b c-a d)^2 \sqrt [3]{a+b x}} \]
[Out]
________________________________________________________________________________________
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 = {47, 37}
\begin {gather*} \frac {9 d \sqrt [3]{c+d x}}{4 \sqrt [3]{a+b x} (b c-a d)^2}-\frac {3 \sqrt [3]{c+d x}}{4 (a+b x)^{4/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx &=-\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}-\frac {(3 d) \int \frac {1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx}{4 (b c-a d)}\\ &=-\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}+\frac {9 d \sqrt [3]{c+d x}}{4 (b c-a d)^2 \sqrt [3]{a+b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 46, normalized size = 0.70 \begin {gather*} \frac {3 \sqrt [3]{c+d x} (-b c+4 a d+3 b d x)}{4 (b c-a d)^2 (a+b x)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 54, normalized size = 0.82
method | result | size |
gosper | \(\frac {3 \left (d x +c \right )^{\frac {1}{3}} \left (3 b d x +4 a d -b c \right )}{4 \left (b x +a \right )^{\frac {4}{3}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (54) = 108\).
time = 0.82, size = 118, normalized size = 1.79 \begin {gather*} \frac {3 \, {\left (3 \, b d x - b c + 4 \, a d\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{4 \, {\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]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {7}{3}} \left (c + d x\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.98, size = 71, normalized size = 1.08 \begin {gather*} \frac {\left (\frac {9\,d\,x}{4\,{\left (a\,d-b\,c\right )}^2}+\frac {12\,a\,d-3\,b\,c}{4\,b\,{\left (a\,d-b\,c\right )}^2}\right )\,{\left (c+d\,x\right )}^{1/3}}{x\,{\left (a+b\,x\right )}^{1/3}+\frac {a\,{\left (a+b\,x\right )}^{1/3}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________