Optimal. Leaf size=278 \[ -\frac {59 \log \left (2 a^{2/3} x^2+2^{2/3} \sqrt [3]{a} x \sqrt [3]{a x^3-b x^2}+\sqrt [3]{2} \left (a x^3-b x^2\right )^{2/3}\right )}{288 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {\left (a x^3-b x^2\right )^{2/3} \left (-625 a^4 x^4-67 a^3 b x^3+1503 a^2 b^2 x^2+91 a b^3 x-1190 b^4\right )}{672 b^4 x (b-a x)^3 (a x+b)^2}+\frac {59 \log \left (2^{2/3} \sqrt [3]{a x^3-b x^2}-2 \sqrt [3]{a} x\right )}{144 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {59 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2^{2/3} \sqrt [3]{a x^3-b x^2}+\sqrt [3]{a} x}\right )}{48 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} b^4} \]
________________________________________________________________________________________
Rubi [C] time = 147.51, antiderivative size = 1306, normalized size of antiderivative = 4.70, number of steps used = 10, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2056, 6733, 6742, 1404, 430, 429}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 429
Rule 430
Rule 1404
Rule 2056
Rule 6733
Rule 6742
Rubi steps
\begin {align*} \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {b^2+a^2 x^2}{x^{2/3} \sqrt [3]{-b+a x} \left (-b^2+a^2 x^2\right )^3} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {b^2+a^2 x^6}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \left (\frac {2 b^2}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^3}+\frac {1}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}+\frac {\left (6 b^2 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-b+a x^3\right )^{7/3} \left (b+a x^3\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}+\frac {\left (6 b^2 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-b+a x^3\right )^{10/3} \left (b+a x^3\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1-\frac {a x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+a x^3\right )^2 \left (1-\frac {a x^3}{b}\right )^{7/3}} \, dx,x,\sqrt [3]{x}\right )}{b^2 \sqrt [3]{-b x^2+a x^3}}-\frac {\left (6 x^{2/3} \sqrt [3]{1-\frac {a x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+a x^3\right )^3 \left (1-\frac {a x^3}{b}\right )^{10/3}} \, dx,x,\sqrt [3]{x}\right )}{b \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {9 x \Gamma \left (\frac {4}{3}\right ) \left (1820 b^4 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )-2275 a b^3 x \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )-585 a^2 b^2 x^2 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )+2457 a^3 b x^3 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )-1053 a^4 x^4 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )+2484 a b^3 x \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )-6534 a^2 b^2 x^2 \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )+5832 a^3 b x^3 \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )-1782 a^4 x^4 \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )+162 a x (7 b-6 a x) (b-a x)^2 \, _3F_2\left (2,2,3;1,\frac {16}{3};\frac {2 a x}{b+a x}\right )+162 a \left (b \sqrt [3]{x}-a x^{4/3}\right )^3 \, _4F_3\left (2,2,2,3;1,1,\frac {16}{3};\frac {2 a x}{b+a x}\right )\right )}{1820 b^4 (b-a x) (b+a x)^3 \sqrt [3]{-b x^2+a x^3} \Gamma \left (\frac {1}{3}\right )}-\frac {9 x \Gamma \left (\frac {4}{3}\right ) \left (553280 b^6 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )-1521520 a b^5 x \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+1482000 a^2 b^4 x^2 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+355680 a^3 b^3 x^3 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )-1723680 a^4 b^2 x^4 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+1200420 a^5 b x^5 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )-277020 a^6 x^6 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+994248 a b^5 x \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )-4219830 a^2 b^4 x^2 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )+7474680 a^3 b^3 x^3 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )-6828300 a^4 b^2 x^4 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )+3178440 a^5 b x^5 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )-599238 a^6 x^6 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )+405 a x (b-a x)^2 \left (1730 b^3-4419 a b^2 x+3960 a^2 b x^2-1215 a^3 x^3\right ) \, _3F_2\left (2,2,4;1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+1215 a \left (b \sqrt [3]{x}-a x^{4/3}\right )^3 \left (191 b^2-336 a b x+153 a^2 x^2\right ) \, _4F_3\left (2,2,2,4;1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+36450 a b^5 x \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-178605 a^2 b^4 x^2 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+349920 a^3 b^3 x^3 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-342630 a^4 b^2 x^4 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+167670 a^5 b x^5 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-32805 a^6 x^6 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+2187 a b^5 x \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-10935 a^2 b^4 x^2 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+21870 a^3 b^3 x^3 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-21870 a^4 b^2 x^4 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+10935 a^5 b x^5 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-2187 a^6 x^6 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )\right )}{276640 b^4 (b-a x)^2 (b+a x)^4 \sqrt [3]{-b x^2+a x^3} \Gamma \left (\frac {1}{3}\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.88, size = 156, normalized size = 0.56 \begin {gather*} \frac {x \sqrt [3]{\frac {a x}{b}+1} \left (-625 a^4 x^4-67 a^3 b x^3+1503 a^2 b^2 x^2+91 a b^3 x-1190 b^4\right )-826 x \sqrt [3]{1-\frac {a x}{b}} \left (b^2-a^2 x^2\right )^2 \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {2 a x}{b+a x}\right )}{672 b^4 (b-a x)^2 \sqrt [3]{x^2 (a x-b)} (a x+b)^2 \sqrt [3]{\frac {a x}{b}+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.24, size = 278, normalized size = 1.00 \begin {gather*} -\frac {59 \log \left (2 a^{2/3} x^2+2^{2/3} \sqrt [3]{a} x \sqrt [3]{a x^3-b x^2}+\sqrt [3]{2} \left (a x^3-b x^2\right )^{2/3}\right )}{288 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {\left (a x^3-b x^2\right )^{2/3} \left (-625 a^4 x^4-67 a^3 b x^3+1503 a^2 b^2 x^2+91 a b^3 x-1190 b^4\right )}{672 b^4 x (b-a x)^3 (a x+b)^2}+\frac {59 \log \left (2^{2/3} \sqrt [3]{a x^3-b x^2}-2 \sqrt [3]{a} x\right )}{144 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {59 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2^{2/3} \sqrt [3]{a x^3-b x^2}+\sqrt [3]{a} x}\right )}{48 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.43, size = 959, normalized size = 3.45 \begin {gather*} \left [\frac {826 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (-\frac {2^{\frac {1}{3}} a^{\frac {1}{3}} x - {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 413 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 2478 \, \sqrt {\frac {1}{6}} {\left (a^{6} x^{6} - a^{5} b x^{5} - 2 \, a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{3} + a^{2} b^{4} x^{2} - a b^{5} x\right )} \sqrt {-\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}} \log \left (-\frac {4 \, a x^{2} - 3 \cdot 2^{\frac {2}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {2}{3}} x - 2 \, b x - 6 \, \sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a^{\frac {4}{3}} x^{2} + {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a x - 2^{\frac {2}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}} a^{\frac {2}{3}}\right )} \sqrt {-\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}}}{a x^{2} + b x}\right ) - 6 \, {\left (625 \, a^{5} x^{4} + 67 \, a^{4} b x^{3} - 1503 \, a^{3} b^{2} x^{2} - 91 \, a^{2} b^{3} x + 1190 \, a b^{4}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{4032 \, {\left (a^{6} b^{4} x^{6} - a^{5} b^{5} x^{5} - 2 \, a^{4} b^{6} x^{4} + 2 \, a^{3} b^{7} x^{3} + a^{2} b^{8} x^{2} - a b^{9} x\right )}}, \frac {826 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (-\frac {2^{\frac {1}{3}} a^{\frac {1}{3}} x - {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 413 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 4956 \, \sqrt {\frac {1}{6}} {\left (a^{6} x^{6} - a^{5} b x^{5} - 2 \, a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{3} + a^{2} b^{4} x^{2} - a b^{5} x\right )} \sqrt {\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}} \arctan \left (\frac {\sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} x + 2 \, {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}}}{x}\right ) - 6 \, {\left (625 \, a^{5} x^{4} + 67 \, a^{4} b x^{3} - 1503 \, a^{3} b^{2} x^{2} - 91 \, a^{2} b^{3} x + 1190 \, a b^{4}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{4032 \, {\left (a^{6} b^{4} x^{6} - a^{5} b^{5} x^{5} - 2 \, a^{4} b^{6} x^{4} + 2 \, a^{3} b^{7} x^{3} + a^{2} b^{8} x^{2} - a b^{9} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{2}+b^{2}}{\left (a^{2} x^{2}-b^{2}\right )^{3} \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx \end {gather*}
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*} \int \frac {a^{2} x^{2} + b^{2}}{{\left (a^{2} x^{2} - b^{2}\right )}^{3} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {a^2\,x^2+b^2}{{\left (b^2-a^2\,x^2\right )}^3\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{2} + b^{2}}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x - b\right )^{3} \left (a x + b\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________