Optimal. Leaf size=134 \[ -\frac {2 d^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac {2 d^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}-\frac {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}} \]
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Rubi [A] time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {47, 63, 331, 298, 205, 208} \begin {gather*} -\frac {2 d^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac {2 d^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}-\frac {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 205
Rule 208
Rule 298
Rule 331
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{(a+b x)^{9/4}} \, dx &=-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac {d \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{5/4}} \, dx}{b}\\ &=-\frac {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac {d^2 \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{b^2}\\ &=-\frac {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac {\left (4 d^2\right ) \operatorname {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 {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac {\left (4 d^2\right ) \operatorname {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 {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac {\left (2 d^{3/2}\right ) \operatorname {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 )}{b^2}-\frac {\left (2 d^{3/2}\right ) \operatorname {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 )}{b^2}\\ &=-\frac {4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac {4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}-\frac {2 d^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac {2 d^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 73, normalized size = 0.54 \begin {gather*} -\frac {4 (c+d x)^{5/4} \, _2F_1\left (-\frac {5}{4},-\frac {5}{4};-\frac {1}{4};\frac {d (a+b x)}{a d-b c}\right )}{5 b (a+b x)^{5/4} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 134, normalized size = 1.00 \begin {gather*} \frac {2 d^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{b^{9/4}}+\frac {2 d^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{b^{9/4}}-\frac {4 \left (\frac {b (c+d x)^{5/4}}{(a+b x)^{5/4}}+\frac {5 d \sqrt [4]{c+d x}}{\sqrt [4]{a+b x}}\right )}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 368, normalized size = 2.75 \begin {gather*} -\frac {20 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac {d^{5}}{b^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} b^{7} d \left (\frac {d^{5}}{b^{9}}\right )^{\frac {3}{4}} - {\left (b^{8} x + a b^{7}\right )} \sqrt {\frac {\sqrt {b x + a} \sqrt {d x + c} d^{2} + {\left (b^{5} x + a b^{4}\right )} \sqrt {\frac {d^{5}}{b^{9}}}}{b x + a}} \left (\frac {d^{5}}{b^{9}}\right )^{\frac {3}{4}}}{b d^{5} x + a d^{5}}\right ) - 5 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac {d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} d + {\left (b^{3} x + a b^{2}\right )} \left (\frac {d^{5}}{b^{9}}\right )^{\frac {1}{4}}}{b x + a}\right ) + 5 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac {d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} d - {\left (b^{3} x + a b^{2}\right )} \left (\frac {d^{5}}{b^{9}}\right )^{\frac {1}{4}}}{b x + a}\right ) + 4 \, {\left (6 \, b d x + b c + 5 \, a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{5 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \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*} \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {9}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{\frac {9}{4}}}\, dx \end {gather*}
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*} \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {9}{4}}}\,{d x} \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 )}^{9/4}} \,d x \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 {9}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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