Optimal. Leaf size=1372 \[ -\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right ) d^{7/3}}{28 \sqrt [3]{2} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {5\ 3^{3/4} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right ),-7-4 \sqrt {3}\right ) d^{7/3}}{7\ 2^{5/6} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {15 \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2} d^{7/3}}{14 \sqrt [3]{2} b^{2/3} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}-\frac {15 (c+d x)^{2/3} d^2}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {15 (c+d x)^{2/3} d}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}} \]
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Rubi [A] time = 2.44, antiderivative size = 1372, 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 = {51, 62, 623, 303, 218, 1877} \[ -\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right ) d^{7/3}}{28 \sqrt [3]{2} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {5\ 3^{3/4} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right ) d^{7/3}}{7\ 2^{5/6} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {15 \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2} d^{7/3}}{14 \sqrt [3]{2} b^{2/3} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}-\frac {15 (c+d x)^{2/3} d^2}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {15 (c+d x)^{2/3} d}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 62
Rule 218
Rule 303
Rule 623
Rule 1877
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}-\frac {(5 d) \int \frac {1}{(a+b x)^{7/3} \sqrt [3]{c+d x}} \, dx}{7 (b c-a d)}\\ &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}+\frac {\left (5 d^2\right ) \int \frac {1}{(a+b x)^{4/3} \sqrt [3]{c+d x}} \, dx}{14 (b c-a d)^2}\\ &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (5 d^3\right ) \int \frac {1}{\sqrt [3]{a+b x} \sqrt [3]{c+d x}} \, dx}{14 (b c-a d)^3}\\ &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (5 d^3 \sqrt [3]{(a+b x) (c+d x)}\right ) \int \frac {1}{\sqrt [3]{a c+(b c+a d) x+b d x^2}} \, dx}{14 (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x}}\\ &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (15 d^3 \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{14 (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (15 d^{8/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} x}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{14\ 2^{2/3} \sqrt [3]{b} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}+\frac {\left (15 d^{8/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{14 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{b} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {15 d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{14 \sqrt [3]{2} b^{2/3} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{28 \sqrt [3]{2} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {5\ 3^{3/4} d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{7\ 2^{5/6} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.05 \[ -\frac {3 \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {7}{3},\frac {1}{3};-\frac {4}{3};\frac {d (a+b x)}{a d-b c}\right )}{7 b (a+b x)^{7/3} \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{b^{4} d x^{5} + a^{4} c + {\left (b^{4} c + 4 \, a b^{3} d\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{2} + {\left (4 \, a^{3} b c + a^{4} d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {10}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {1}{\left (b x +a \right )^{\frac {10}{3}} \left (d x +c \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {10}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,x\right )}^{10/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {10}{3}} \sqrt [3]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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