Optimal. Leaf size=97 \[ -\frac{208 a^3 \sqrt{1-a x}}{105 \sqrt{a x}}-\frac{104 a^3 \sqrt{1-a x}}{105 (a x)^{3/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0266511, antiderivative size = 97, normalized size of antiderivative = 1., 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 = {16, 78, 45, 37} \[ -\frac{208 a^3 \sqrt{1-a x}}{105 \sqrt{a x}}-\frac{104 a^3 \sqrt{1-a x}}{105 (a x)^{3/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1+a x}{x^4 \sqrt{a x} \sqrt{1-a x}} \, dx &=a^4 \int \frac{1+a x}{(a x)^{9/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}}+\frac{1}{7} \left (13 a^4\right ) \int \frac{1}{(a x)^{7/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}+\frac{1}{35} \left (52 a^4\right ) \int \frac{1}{(a x)^{5/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}-\frac{104 a^3 \sqrt{1-a x}}{105 (a x)^{3/2}}+\frac{1}{105} \left (104 a^4\right ) \int \frac{1}{(a x)^{3/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}-\frac{104 a^3 \sqrt{1-a x}}{105 (a x)^{3/2}}-\frac{208 a^3 \sqrt{1-a x}}{105 \sqrt{a x}}\\ \end{align*}
Mathematica [A] time = 0.0166348, size = 45, normalized size = 0.46 \[ -\frac{2 \sqrt{-a x (a x-1)} \left (104 a^3 x^3+52 a^2 x^2+39 a x+15\right )}{105 a x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 41, normalized size = 0.4 \begin{align*} -{\frac{208\,{a}^{3}{x}^{3}+104\,{a}^{2}{x}^{2}+78\,ax+30}{105\,{x}^{3}}\sqrt{-ax+1}{\frac{1}{\sqrt{ax}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44396, size = 111, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt{a x} \sqrt{-a x + 1}}{105 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 18.8721, size = 274, normalized size = 2.82 \begin{align*} a \left (\begin{cases} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{15} - \frac{8 a \sqrt{-1 + \frac{1}{a x}}}{15 x} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{5 x^{2}} & \text{for}\: \frac{1}{\left |{a x}\right |} > 1 \\- \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{15} - \frac{8 i a \sqrt{1 - \frac{1}{a x}}}{15 x} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{5 x^{2}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{32 a^{3} \sqrt{-1 + \frac{1}{a x}}}{35} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{35 x} - \frac{12 a \sqrt{-1 + \frac{1}{a x}}}{35 x^{2}} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{7 x^{3}} & \text{for}\: \frac{1}{\left |{a x}\right |} > 1 \\- \frac{32 i a^{3} \sqrt{1 - \frac{1}{a x}}}{35} - \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{35 x} - \frac{12 i a \sqrt{1 - \frac{1}{a x}}}{35 x^{2}} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{7 x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.89268, size = 236, normalized size = 2.43 \begin{align*} -\frac{\frac{15 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac{7}{2}}} + \frac{231 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac{5}{2}}} + \frac{1435 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac{3}{2}}} + \frac{7875 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}}{\sqrt{a x}} - \frac{{\left (15 \, a^{4} + \frac{231 \, a^{3}{\left (\sqrt{-a x + 1} - 1\right )}^{2}}{x} + \frac{1435 \, a^{2}{\left (\sqrt{-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac{7875 \, a{\left (\sqrt{-a x + 1} - 1\right )}^{6}}{x^{3}}\right )} \left (a x\right )^{\frac{7}{2}}}{{\left (\sqrt{-a x + 1} - 1\right )}^{7}}}{6720 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]