Optimal. Leaf size=457 \[ -\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),4 \sqrt {3}-7\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {108 \sqrt {a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{935 b d^2}+\frac {12 (a+b x)^{3/2} \sqrt [3]{c+d x} (b c-a d)}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b} \]
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Rubi [A] time = 0.63, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 63, 219} \[ -\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {108 \sqrt {a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{935 b d^2}+\frac {12 (a+b x)^{3/2} \sqrt [3]{c+d x} (b c-a d)}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 219
Rubi steps
\begin {align*} \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx &=\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}+\frac {(2 (b c-a d)) \int \frac {(a+b x)^{3/2}}{(c+d x)^{2/3}} \, dx}{17 b}\\ &=\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}-\frac {\left (18 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{2/3}} \, dx}{187 b d}\\ &=-\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}+\frac {\left (54 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx}{935 b d^2}\\ &=-\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}+\frac {\left (162 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{935 b d^3}\\ &=-\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}-\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \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 (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 73, normalized size = 0.16 \[ \frac {2 (a+b x)^{5/2} \sqrt [3]{c+d x} \, _2F_1\left (-\frac {1}{3},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{a d-b c}\right )}{5 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {1}{3}}\, dx \]
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 \[ \int {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}}\,{d x} \]
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 \[ \int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{1/3} \,d x \]
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 \[ \int \left (a + b x\right )^{\frac {3}{2}} \sqrt [3]{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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