Optimal. Leaf size=192 \[ \frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )-\frac {2 a c d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3}}+\frac {a c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}-\frac {a c d \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1867, 1900,
267, 1907, 252, 251, 337} \begin {gather*} -\frac {2 a c d \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3}}-\frac {a c d \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{2/3}}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+\frac {a c^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}+\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 251
Rule 252
Rule 267
Rule 337
Rule 1867
Rule 1900
Rule 1907
Rubi steps
\begin {align*} \int (c+d x)^2 \sqrt [3]{a+b x^3} \, dx &=\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+a \int \frac {\frac {c^2}{2}+\frac {2 c d x}{3}+\frac {d^2 x^2}{4}}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+a \int \frac {\frac {c^2}{2}+\frac {2 c d x}{3}}{\left (a+b x^3\right )^{2/3}} \, dx+\frac {1}{4} \left (a d^2\right ) \int \frac {x^2}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+a \int \left (\frac {c^2}{2 \left (a+b x^3\right )^{2/3}}+\frac {2 c d x}{3 \left (a+b x^3\right )^{2/3}}\right ) \, dx\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+\frac {1}{2} \left (a c^2\right ) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx+\frac {1}{3} (2 a c d) \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+\frac {1}{3} (2 a c d) \text {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {\left (a c^2 \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{2 \left (a+b x^3\right )^{2/3}}\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+\frac {a c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}+\frac {(2 a c d) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 \sqrt [3]{b}}-\frac {(2 a c d) \text {Subst}\left (\int \frac {1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 \sqrt [3]{b}}\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+\frac {a c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}-\frac {2 a c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}+\frac {(a c d) \text {Subst}\left (\int \frac {\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}-\frac {(a c d) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )+\frac {a c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}-\frac {2 a c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}+\frac {a c d \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}+\frac {(2 a c d) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}\\ &=\frac {a d^2 \sqrt [3]{a+b x^3}}{4 b}+\frac {1}{12} \sqrt [3]{a+b x^3} \left (6 c^2 x+8 c d x^2+3 d^2 x^3\right )-\frac {2 a c d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3}}+\frac {a c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}}-\frac {2 a c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}+\frac {a c d \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.86, size = 111, normalized size = 0.58 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (4 b c^2 x \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+d \left (d \left (a+b x^3\right ) \sqrt [3]{1+\frac {b x^3}{a}}+4 b c x^2 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )\right )}{4 b \sqrt [3]{1+\frac {b x^3}{a}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.59, size = 114, normalized size = 0.59 \begin {gather*} \frac {\sqrt [3]{a} c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {2 \sqrt [3]{a} c d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + d^{2} \left (\begin {cases} \frac {\sqrt [3]{a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {\left (a + b x^{3}\right )^{\frac {4}{3}}}{4 b} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,x^3+a\right )}^{1/3}\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________