Optimal. Leaf size=100 \[ \frac {32 c^3 \sqrt {b x+c x^2}}{35 b^4 x}-\frac {16 c^2 \sqrt {b x+c x^2}}{35 b^3 x^2}+\frac {12 c \sqrt {b x+c x^2}}{35 b^2 x^3}-\frac {2 \sqrt {b x+c x^2}}{7 b x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac {32 c^3 \sqrt {b x+c x^2}}{35 b^4 x}-\frac {16 c^2 \sqrt {b x+c x^2}}{35 b^3 x^2}+\frac {12 c \sqrt {b x+c x^2}}{35 b^2 x^3}-\frac {2 \sqrt {b x+c x^2}}{7 b x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 650
Rule 658
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {b x+c x^2}} \, dx &=-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}-\frac {(6 c) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{7 b}\\ &=-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}+\frac {12 c \sqrt {b x+c x^2}}{35 b^2 x^3}+\frac {\left (24 c^2\right ) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{35 b^2}\\ &=-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}+\frac {12 c \sqrt {b x+c x^2}}{35 b^2 x^3}-\frac {16 c^2 \sqrt {b x+c x^2}}{35 b^3 x^2}-\frac {\left (16 c^3\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{35 b^3}\\ &=-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}+\frac {12 c \sqrt {b x+c x^2}}{35 b^2 x^3}-\frac {16 c^2 \sqrt {b x+c x^2}}{35 b^3 x^2}+\frac {32 c^3 \sqrt {b x+c x^2}}{35 b^4 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 51, normalized size = 0.51 \[ \frac {2 \sqrt {x (b+c x)} \left (-5 b^3+6 b^2 c x-8 b c^2 x^2+16 c^3 x^3\right )}{35 b^4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 49, normalized size = 0.49 \[ \frac {2 \, {\left (16 \, c^{3} x^{3} - 8 \, b c^{2} x^{2} + 6 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt {c x^{2} + b x}}{35 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 107, normalized size = 1.07 \[ \frac {2 \, {\left (70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{\frac {3}{2}} + 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} \sqrt {c} + 5 \, b^{3}\right )}}{35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 55, normalized size = 0.55 \[ -\frac {2 \left (c x +b \right ) \left (-16 x^{3} c^{3}+8 b \,x^{2} c^{2}-6 b^{2} x c +5 b^{3}\right )}{35 \sqrt {c \,x^{2}+b x}\, b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 84, normalized size = 0.84 \[ \frac {32 \, \sqrt {c x^{2} + b x} c^{3}}{35 \, b^{4} x} - \frac {16 \, \sqrt {c x^{2} + b x} c^{2}}{35 \, b^{3} x^{2}} + \frac {12 \, \sqrt {c x^{2} + b x} c}{35 \, b^{2} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x}}{7 \, b x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.20, size = 84, normalized size = 0.84 \[ \frac {32\,c^3\,\sqrt {c\,x^2+b\,x}}{35\,b^4\,x}-\frac {16\,c^2\,\sqrt {c\,x^2+b\,x}}{35\,b^3\,x^2}-\frac {2\,\sqrt {c\,x^2+b\,x}}{7\,b\,x^4}+\frac {12\,c\,\sqrt {c\,x^2+b\,x}}{35\,b^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \sqrt {x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________