Optimal. Leaf size=197 \[ -\frac {693 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{64 a^{13/2}}+\frac {693 b^4 \sqrt {a x+b \sqrt {x}}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{32 a^5}+\frac {231 b^2 x \sqrt {a x+b \sqrt {x}}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {a x+b \sqrt {x}}}{20 a^3}+\frac {22 x^2 \sqrt {a x+b \sqrt {x}}}{5 a^2}-\frac {4 x^3}{a \sqrt {a x+b \sqrt {x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2018, 668, 670, 640, 620, 206} \[ \frac {693 b^4 \sqrt {a x+b \sqrt {x}}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{32 a^5}+\frac {231 b^2 x \sqrt {a x+b \sqrt {x}}}{40 a^4}-\frac {693 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{64 a^{13/2}}-\frac {99 b x^{3/2} \sqrt {a x+b \sqrt {x}}}{20 a^3}+\frac {22 x^2 \sqrt {a x+b \sqrt {x}}}{5 a^2}-\frac {4 x^3}{a \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 620
Rule 640
Rule 668
Rule 670
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^7}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {22 \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {(99 b) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{5 a^2}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}+\frac {\left (693 b^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{40 a^3}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {\left (231 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{16 a^4}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}+\frac {\left (693 b^4\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^5}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {\left (693 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{128 a^6}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {\left (693 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^6}\\ &=-\frac {4 x^3}{a \sqrt {b \sqrt {x}+a x}}+\frac {693 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^6}-\frac {231 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {231 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^4}-\frac {99 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^3}+\frac {22 x^2 \sqrt {b \sqrt {x}+a x}}{5 a^2}-\frac {693 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^{13/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 64, normalized size = 0.32 \[ \frac {4 x^{7/2} \sqrt {\frac {a \sqrt {x}}{b}+1} \, _2F_1\left (\frac {3}{2},\frac {13}{2};\frac {15}{2};-\frac {a \sqrt {x}}{b}\right )}{13 b \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 549, normalized size = 2.79 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (-4480 a^{3} b^{5} x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+1015 a^{3} b^{5} x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-8960 a^{2} b^{6} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2030 a^{2} b^{6} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-4060 \sqrt {a x +b \sqrt {x}}\, a^{\frac {9}{2}} b^{3} x^{\frac {3}{2}}-4480 a \,b^{7} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+1015 a \,b^{7} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+256 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{2}-10150 \sqrt {a x +b \sqrt {x}}\, a^{\frac {7}{2}} b^{4} x +8960 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {7}{2}} b^{4} x -352 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{\frac {3}{2}}-8120 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b^{5} \sqrt {x}+17920 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} b^{5} \sqrt {x}+528 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x -2030 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b^{6}+8960 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b^{6}+3136 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} \sqrt {x}+2000 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4}-2560 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4}\right )}{640 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} a^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________