Optimal. Leaf size=167 \[ -\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {21 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}+\frac {21 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}} \]
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Rubi [A]
time = 0.07, antiderivative size = 167, 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 = {52, 65, 338,
304, 211, 214} \begin {gather*} -\frac {21 (b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}+\frac {21 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}-\frac {7 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 211
Rule 214
Rule 304
Rule 338
Rubi steps
\begin {align*} \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx &=\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {(7 (b c-a d)) \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx}{8 d}\\ &=-\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}+\frac {\left (21 (b c-a d)^2\right ) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{32 d^2}\\ &=-\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}+\frac {\left (21 (b c-a d)^2\right ) \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 )}{8 b d^2}\\ &=-\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}+\frac {\left (21 (b c-a d)^2\right ) \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 )}{8 b d^2}\\ &=-\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}+\frac {\left (21 (b c-a d)^2\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 )}{16 d^{5/2}}-\frac {\left (21 (b c-a d)^2\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 )}{16 d^{5/2}}\\ &=-\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {21 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}+\frac {21 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 143, normalized size = 0.86 \begin {gather*} \frac {2 d^{3/4} (a+b x)^{3/4} \sqrt [4]{c+d x} (-7 b c+11 a d+4 b d x)+\frac {21 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{\sqrt [4]{b}}+\frac {21 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{\sqrt [4]{b}}}{16 d^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {7}{4}}}{\left (d x +c \right )^{\frac {3}{4}}}\, dx\]
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. 1457 vs.
\(2 (127) = 254\).
time = 1.01, size = 1457, normalized size = 8.72 \begin {gather*} -\frac {84 \, d^{2} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b^{3} c^{2} d^{8} - 2 \, a b^{2} c d^{9} + a^{2} b d^{10}\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {3}{4}} - {\left (b^{2} d^{8} x + a b d^{8}\right )} \sqrt {\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (b d^{6} x + a d^{6}\right )} \sqrt {\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}}}{b x + a}} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {3}{4}}}{a b^{8} c^{8} - 8 \, a^{2} b^{7} c^{7} d + 28 \, a^{3} b^{6} c^{6} d^{2} - 56 \, a^{4} b^{5} c^{5} d^{3} + 70 \, a^{5} b^{4} c^{4} d^{4} - 56 \, a^{6} b^{3} c^{3} d^{5} + 28 \, a^{7} b^{2} c^{2} d^{6} - 8 \, a^{8} b c d^{7} + a^{9} d^{8} + {\left (b^{9} c^{8} - 8 \, a b^{8} c^{7} d + 28 \, a^{2} b^{7} c^{6} d^{2} - 56 \, a^{3} b^{6} c^{5} d^{3} + 70 \, a^{4} b^{5} c^{4} d^{4} - 56 \, a^{5} b^{4} c^{3} d^{5} + 28 \, a^{6} b^{3} c^{2} d^{6} - 8 \, a^{7} b^{2} c d^{7} + a^{8} b d^{8}\right )} x}\right ) - 21 \, d^{2} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {1}{4}} \log \left (\frac {21 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (b d^{3} x + a d^{3}\right )} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 21 \, d^{2} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {1}{4}} \log \left (\frac {21 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} - {\left (b d^{3} x + a d^{3}\right )} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b d^{11}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 4 \, {\left (4 \, b d x - 7 \, b c + 11 \, a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{32 \, d^{2}} \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 (a + b x\right )^{\frac {7}{4}}}{\left (c + d x\right )^{\frac {3}{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 (a+b\,x\right )}^{7/4}}{{\left (c+d\,x\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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