Optimal. Leaf size=205 \[ \frac {5 (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}+\frac {5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}{96 b^2 d}+\frac {5 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b} \]
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Rubi [A] time = 0.14, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {50, 63, 331, 298, 205, 208} \begin {gather*} \frac {5 (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}+\frac {5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}{96 b^2 d}+\frac {5 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rule 208
Rule 298
Rule 331
Rubi steps
\begin {align*} \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx &=\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac {(5 (b c-a d)) \int (a+b x)^{3/4} \sqrt [4]{c+d x} \, dx}{12 b}\\ &=\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx}{96 b^2}\\ &=\frac {5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac {\left (5 (b c-a d)^3\right ) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{128 b^2 d}\\ &=\frac {5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac {\left (5 (b c-a d)^3\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 )}{32 b^3 d}\\ &=\frac {5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac {\left (5 (b c-a d)^3\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 )}{32 b^3 d}\\ &=\frac {5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac {\left (5 (b c-a d)^3\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 )}{64 b^2 d^{3/2}}+\frac {\left (5 (b c-a d)^3\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 )}{64 b^2 d^{3/2}}\\ &=\frac {5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac {5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac {(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac {5 (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 73, normalized size = 0.36 \begin {gather*} \frac {4 (a+b x)^{7/4} (c+d x)^{5/4} \, _2F_1\left (-\frac {5}{4},\frac {7}{4};\frac {11}{4};\frac {d (a+b x)}{a d-b c}\right )}{7 b \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 = 0.53, size = 218, normalized size = 1.06 \begin {gather*} -\frac {5 (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{64 b^{9/4} d^{7/4}}-\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{64 b^{9/4} d^{7/4}}+\frac {(b c-a d)^3 \left (\frac {5 b^2 (c+d x)^{9/4}}{(a+b x)^{9/4}}-\frac {15 d^2 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x}}+\frac {42 b d (c+d x)^{5/4}}{(a+b x)^{5/4}}\right )}{96 b^2 d \left (\frac {b (c+d x)}{a+b x}-d\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.56, size = 2151, normalized size = 10.49
result too large to display
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 {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {5}{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {5}{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 {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {5}{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 = -0.00 \begin {gather*} \int {\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{5/4} \,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 \left (a + b x\right )^{\frac {3}{4}} \left (c + d x\right )^{\frac {5}{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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