Optimal. Leaf size=152 \[ -\frac {5 \sqrt [4]{b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac {5 \sqrt [4]{b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]
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Rubi [A] time = 0.09, 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 = {47, 50, 63, 240, 212, 208, 205} \[ \frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {5 \sqrt [4]{b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac {5 \sqrt [4]{b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rule 208
Rule 212
Rule 240
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx &=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {(5 b) \int \frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx}{d}\\ &=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {(5 b (b c-a d)) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 d^2}\\ &=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{d^2}\\ &=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{d^2}\\ &=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {\left (5 \sqrt {b} (b c-a d)\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 )}{2 d^2}-\frac {\left (5 \sqrt {b} (b c-a d)\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 )}{2 d^2}\\ &=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {5 \sqrt [4]{b} (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 d^{9/4}}-\frac {5 \sqrt [4]{b} (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 d^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 73, normalized size = 0.48 \[ \frac {4 (a+b x)^{9/4} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac {5}{4},\frac {9}{4};\frac {13}{4};\frac {d (a+b x)}{a d-b c}\right )}{9 b (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 857, normalized size = 5.64 \[ -\frac {20 \, {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (b c d^{7} - a d^{8}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {3}{4}} + {\left (d^{8} x + c d^{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 (d^{5} x + c d^{4}\right )} \sqrt {\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}}}{d x + c}} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {3}{4}}}{b^{5} c^{5} - 4 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + {\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 (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 5 \, {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 4 \, {\left (b d x + 5 \, b c - 4 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, {\left (d^{3} x + c d^{2}\right )}} \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {5}{4}}}{\left (d x +c \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {5}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{5/4}}{{\left (c+d\,x\right )}^{5/4}} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {5}{4}}}{\left (c + d x\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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