Optimal. Leaf size=116 \[ \frac {32 b^3 \sqrt {a x^2+b x^3}}{35 a^4 x^{3/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \begin {gather*} \frac {32 b^3 \sqrt {a x^2+b x^3}}{35 a^4 x^{3/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^3}} \, dx &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}-\frac {(6 b) \int \frac {1}{x^{5/2} \sqrt {a x^2+b x^3}} \, dx}{7 a}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}+\frac {\left (24 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^3}} \, dx}{35 a^2}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}-\frac {\left (16 b^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx}{35 a^3}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}+\frac {32 b^3 \sqrt {a x^2+b x^3}}{35 a^4 x^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 55, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {x^2 (a+b x)} \left (-5 a^3+6 a^2 b x-8 a b^2 x^2+16 b^3 x^3\right )}{35 a^4 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 57, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {a x^2+b x^3} \left (-5 a^3+6 a^2 b x-8 a b^2 x^2+16 b^3 x^3\right )}{35 a^4 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 51, normalized size = 0.44 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{35 \, a^{4} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 103, normalized size = 0.89 \begin {gather*} \frac {64 \, {\left (35 \, {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{6} - 21 \, a {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{4} + 7 \, a^{2} {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a^{3}\right )} b^{\frac {7}{2}}}{35 \, {\left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 57, normalized size = 0.49 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 \sqrt {b \,x^{3}+a \,x^{2}}\, a^{4} x^{\frac {5}{2}}} \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 {1}{\sqrt {b x^{3} + a x^{2}} x^{\frac {7}{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 {1}{x^{7/2}\,\sqrt {b\,x^3+a\,x^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 {1}{x^{\frac {7}{2}} \sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________