Optimal. Leaf size=152 \[ \frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}-\frac {5 \sqrt [4]{d} (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}} \]
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Rubi [A]
time = 0.06, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {49, 52, 65, 338,
304, 211, 214} \begin {gather*} -\frac {5 \sqrt [4]{d} (b c-a d) \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 211
Rule 214
Rule 304
Rule 338
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx &=-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d) \int \frac {\sqrt [4]{c+d x}}{\sqrt [4]{a+b x}} \, dx}{b}\\ &=\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d (b c-a d)) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{4 b^2}\\ &=\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d (b c-a d)) \text {Subst}\left (\int \frac {x^2}{\left (c-\frac {a d}{b}+\frac {d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{b^3}\\ &=\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d (b c-a d)) \text {Subst}\left (\int \frac {x^2}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{b^3}\\ &=\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {\left (5 \sqrt {d} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 b^2}-\frac {\left (5 \sqrt {d} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 b^2}\\ &=\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}-\frac {5 \sqrt [4]{d} (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.05, size = 71, normalized size = 0.47 \begin {gather*} -\frac {4 (c+d x)^{5/4} \, _2F_1\left (-\frac {5}{4},-\frac {1}{4};\frac {3}{4};\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [4]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{\frac {5}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs.
\(2 (116) = 232\).
time = 1.50, size = 857, normalized size = 5.64 \begin {gather*} \frac {20 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (b^{8} c - a b^{7} d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {3}{4}} + {\left (b^{8} x + a b^{7}\right )} \sqrt {\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (b^{5} x + a b^{4}\right )} \sqrt {\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}}}{b x + a}} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {3}{4}}}{a b^{4} c^{4} d - 4 \, a^{2} b^{3} c^{3} d^{2} + 6 \, a^{3} b^{2} c^{2} d^{3} - 4 \, a^{4} b c d^{4} + a^{5} d^{5} + {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}\right ) + 5 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 5 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} - {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 4 \, {\left (b d x - 4 \, b c + 5 \, a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{4 \, {\left (b^{3} x + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {5}{4}}}{\left (a + b x\right )^{\frac {5}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/4}}{{\left (a+b\,x\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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