Optimal. Leaf size=320 \[ -\frac {16 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (14 A b-5 a B) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {16 \sqrt {e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt {a+b x^3}}-\frac {2 \sqrt {e x} (14 A b-5 a B)}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {453, 290, 329, 225} \[ -\frac {16 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (14 A b-5 a B) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {16 \sqrt {e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt {a+b x^3}}-\frac {2 \sqrt {e x} (14 A b-5 a B)}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 225
Rule 290
Rule 329
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^3}{(e x)^{7/2} \left (a+b x^3\right )^{5/2}} \, dx &=-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac {(14 A b-5 a B) \int \frac {1}{\sqrt {e x} \left (a+b x^3\right )^{5/2}} \, dx}{5 a e^3}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac {2 (14 A b-5 a B) \sqrt {e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {(8 (14 A b-5 a B)) \int \frac {1}{\sqrt {e x} \left (a+b x^3\right )^{3/2}} \, dx}{45 a^2 e^3}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac {2 (14 A b-5 a B) \sqrt {e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {16 (14 A b-5 a B) \sqrt {e x}}{135 a^3 e^4 \sqrt {a+b x^3}}-\frac {(16 (14 A b-5 a B)) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^3}} \, dx}{135 a^3 e^3}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac {2 (14 A b-5 a B) \sqrt {e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {16 (14 A b-5 a B) \sqrt {e x}}{135 a^3 e^4 \sqrt {a+b x^3}}-\frac {(32 (14 A b-5 a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{135 a^3 e^4}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac {2 (14 A b-5 a B) \sqrt {e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac {16 (14 A b-5 a B) \sqrt {e x}}{135 a^3 e^4 \sqrt {a+b x^3}}-\frac {16 (14 A b-5 a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.09, size = 121, normalized size = 0.38 \[ \frac {x \left (a^2 \left (110 B x^3-54 A\right )+32 x^3 \left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} (5 a B-14 A b) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\frac {b x^3}{a}\right )+a \left (80 b B x^6-308 A b x^3\right )-224 A b^2 x^6\right )}{135 a^3 (e x)^{7/2} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{b^{3} e^{4} x^{13} + 3 \, a b^{2} e^{4} x^{10} + 3 \, a^{2} b e^{4} x^{7} + a^{3} e^{4} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.06, size = 7299, normalized size = 22.81 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {B\,x^3+A}{{\left (e\,x\right )}^{7/2}\,{\left (b\,x^3+a\right )}^{5/2}} \,d x \]
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]
________________________________________________________________________________________