Optimal. Leaf size=122 \[ \frac {2 a^{3/2} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}}-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2028, 2031, 2029, 206} \begin {gather*} \frac {2 a^{3/2} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}}-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx &=-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {a \int \frac {\sqrt {a x+b x^n}}{(c x)^{3/2}} \, dx}{c}\\ &=-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {a^2 \int \frac {1}{\sqrt {c x} \sqrt {a x+b x^n}} \, dx}{c^2}\\ &=-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {\left (a^2 \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a x+b x^n}} \, dx}{c^2 \sqrt {c x}}\\ &=-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {\left (2 a^2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}}\\ &=-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {2 a^{3/2} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 120, normalized size = 0.98 \begin {gather*} \frac {x \left (-6 a^{3/2} \sqrt {b} x^{\frac {n+3}{2}} \sqrt {\frac {a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {1}{2}-\frac {n}{2}}}{\sqrt {b}}\right )+8 a^2 x^2+10 a b x^{n+1}+2 b^2 x^{2 n}\right )}{3 (n-1) (c x)^{5/2} \sqrt {a x+b x^n}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 2.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x + b x^{n}\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x +b \,x^{n}\right )^{\frac {3}{2}}}{\left (c x \right )^{\frac {5}{2}}}\, 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 {{\left (a x + b x^{n}\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {5}{2}}}\,{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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^n+a\,x\right )}^{3/2}}{{\left (c\,x\right )}^{5/2}} \,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 {\left (a x + b x^{n}\right )^{\frac {3}{2}}}{\left (c x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________