Optimal. Leaf size=471 \[ -\frac {4807 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 b^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 b^{27/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {4807 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 b^7 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 a^5 \sqrt {a x+b \sqrt [3]{x}}}{221 b^7 \sqrt [3]{x}}-\frac {4807 a^4 \sqrt {a x+b \sqrt [3]{x}}}{663 b^6 x}+\frac {24035 a^3 \sqrt {a x+b \sqrt [3]{x}}}{4641 b^5 x^{5/3}}-\frac {6555 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1547 b^4 x^{7/3}}+\frac {437 a \sqrt {a x+b \sqrt [3]{x}}}{119 b^3 x^3}-\frac {23 \sqrt {a x+b \sqrt [3]{x}}}{7 b^2 x^{11/3}}+\frac {3}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.69, antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2018, 2023, 2025, 2032, 329, 305, 220, 1196} \[ -\frac {4807 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 b^7 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {24035 a^3 \sqrt {a x+b \sqrt [3]{x}}}{4641 b^5 x^{5/3}}-\frac {6555 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1547 b^4 x^{7/3}}-\frac {4807 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 b^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 b^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 a^5 \sqrt {a x+b \sqrt [3]{x}}}{221 b^7 \sqrt [3]{x}}-\frac {4807 a^4 \sqrt {a x+b \sqrt [3]{x}}}{663 b^6 x}+\frac {437 a \sqrt {a x+b \sqrt [3]{x}}}{119 b^3 x^3}-\frac {23 \sqrt {a x+b \sqrt [3]{x}}}{7 b^2 x^{11/3}}+\frac {3}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2023
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^{10} \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {69 \operatorname {Subst}\left (\int \frac {1}{x^{11} \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}-\frac {(437 a) \operatorname {Subst}\left (\int \frac {1}{x^9 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14 b^2}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}+\frac {\left (6555 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^7 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{238 b^3}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}-\frac {\left (72105 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3094 b^4}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}+\frac {\left (24035 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1326 b^5}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}-\frac {\left (4807 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 b^6}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^6\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 b^7}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^6 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{442 b^7 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^6 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 b^7 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 b^{13/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (4807 a^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 b^{13/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{11/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 b^7 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}+\frac {4807 a^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.08, size = 64, normalized size = 0.14 \[ -\frac {2 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (-\frac {21}{4},\frac {3}{2};-\frac {17}{4};-\frac {a x^{2/3}}{b}\right )}{7 b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 7.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{3} + 3 \, a^{2} b^{2} x^{\frac {5}{3}} - 2 \, a b^{3} x - {\left (2 \, a^{3} b x^{2} - b^{4}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{6} x^{9} + 2 \, a^{3} b^{3} x^{7} + b^{6} x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 411, normalized size = 0.87 \[ \frac {-201894 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+100947 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+201894 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{6} x^{\frac {22}{3}}+174048 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}}-39452 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}}-19228 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{4} b^{2} x^{6}+8740 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b^{3} x^{\frac {16}{3}}-5244 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b^{4} x^{\frac {14}{3}}+3588 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{5} x^{4}-2652 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{6} x^{\frac {10}{3}}}{9282 \left (a \,x^{\frac {2}{3}}+b \right ) b^{7} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________