Optimal. Leaf size=896 \[ -\frac {27 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}-\frac {81 \left (1+\sqrt {3}\right ) (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{448 b^{5/3} d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac {81 \sqrt [4]{3} (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{5/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {27\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{5/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Rubi [A]
time = 0.75, antiderivative size = 896, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {52, 65, 314,
231, 1895} \begin {gather*} -\frac {81 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\text {ArcCos}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{448 b^{5/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {27\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\text {ArcCos}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{896 b^{5/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {81 \left (1+\sqrt {3}\right ) \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{5/3} d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac {27 \sqrt {a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 b d^2}+\frac {3 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 231
Rule 314
Rule 1895
Rubi steps
\begin {align*} \int (a+b x)^{3/2} (c+d x)^{5/6} \, dx &=\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}+\frac {(b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt [6]{c+d x}} \, dx}{4 b}\\ &=\frac {3 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}-\frac {\left (9 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt [6]{c+d x}} \, dx}{56 b d}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}+\frac {\left (27 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}} \, dx}{448 b d^2}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}+\frac {\left (81 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{224 b d^3}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}-\frac {\left (81 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) (b c-a d)^{2/3}-2 b^{2/3} x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{5/3} d^3}-\frac {\left (81 \left (1-\sqrt {3}\right ) (b c-a d)^{11/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{5/3} d^3}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 b d}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 b}-\frac {81 \left (1+\sqrt {3}\right ) (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{448 b^{5/3} d^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac {81 \sqrt [4]{3} (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{5/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {27\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{5/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.08, size = 73, normalized size = 0.08 \begin {gather*} \frac {2 (a+b x)^{5/2} (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{-b c+a d}\right )}{5 b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{6}}\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{6}}\, 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 = -0.00 \begin {gather*} \int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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