Optimal. Leaf size=438 \[ -\frac {44 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {88 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 a^5 \sqrt {a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac {88 a^4 \sqrt {a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac {88 a^3 \sqrt {a x+b \sqrt [3]{x}}}{4641 b^2 x^{5/3}}-\frac {24 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1547 b x^{7/3}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{7 x^4}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{119 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.64, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2018, 2020, 2025, 2032, 329, 305, 220, 1196} \[ -\frac {88 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 a^3 \sqrt {a x+b \sqrt [3]{x}}}{4641 b^2 x^{5/3}}-\frac {44 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 a^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 a^5 \sqrt {a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac {88 a^4 \sqrt {a x+b \sqrt [3]{x}}}{3315 b^3 x}-\frac {24 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1547 b x^{7/3}}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{119 x^3}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{7 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2020
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^5} \, dx &=3 \operatorname {Subst}\left (\int \frac {\left (b x+a x^3\right )^{3/2}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}+\frac {1}{7} (6 a) \operatorname {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}+\frac {1}{119} \left (12 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^7 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}-\frac {\left (132 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1547 b}\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}+\frac {\left (44 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 b^2}\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {88 a^4 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}-\frac {\left (44 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^3}\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {88 a^4 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {88 a^5 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}-\frac {\left (44 a^6\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^4}\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {88 a^4 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {88 a^5 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}-\frac {\left (44 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 )}{1105 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {88 a^4 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {88 a^5 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}-\frac {\left (88 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 )}{1105 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {88 a^4 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {88 a^5 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}-\frac {\left (88 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 )}{1105 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (88 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 )}{1105 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {88 a^{11/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{119 x^3}-\frac {24 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b x^{7/3}}+\frac {88 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^2 x^{5/3}}-\frac {88 a^4 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {88 a^5 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{7 x^4}+\frac {88 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 )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {44 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 )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 62, normalized size = 0.14 \[ -\frac {2 b \sqrt {a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac {21}{4},-\frac {3}{2};-\frac {17}{4};-\frac {a x^{2/3}}{b}\right )}{7 x^{11/3} \sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 8.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 411, normalized size = 0.94 \[ \frac {-\frac {88 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {44 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {88 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{6} x^{\frac {22}{3}}}{1105}+\frac {88 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}}}{1105}-\frac {88 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {20}{3}}}{3315}-\frac {176 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{4} b^{2} x^{6}}{23205}+\frac {16 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b^{3} x^{\frac {16}{3}}}{4641}-\frac {622 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b^{4} x^{\frac {14}{3}}}{1547}-\frac {80 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{5} x^{4}}{119}-\frac {2 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{6} x^{\frac {10}{3}}}{7}}{\left (a \,x^{\frac {2}{3}}+b \right ) b^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________