Optimal. Leaf size=152 \[ \frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}+\frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{a \left (a+c x^2\right )^{3/2} (64 a B-105 A c x)}{840 c^3}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.128524, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {833, 780, 195, 217, 206} \[ \frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}+\frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{a \left (a+c x^2\right )^{3/2} (64 a B-105 A c x)}{840 c^3}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 (A+B x) \sqrt{a+c x^2} \, dx &=\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{\int x^3 (-4 a B+7 A c x) \sqrt{a+c x^2} \, dx}{7 c}\\ &=\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{\int x^2 (-21 a A c-24 a B c x) \sqrt{a+c x^2} \, dx}{42 c^2}\\ &=-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{\int x \left (48 a^2 B c-105 a A c^2 x\right ) \sqrt{a+c x^2} \, dx}{210 c^3}\\ &=-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{a (64 a B-105 A c x) \left (a+c x^2\right )^{3/2}}{840 c^3}+\frac{\left (a^2 A\right ) \int \sqrt{a+c x^2} \, dx}{8 c^2}\\ &=\frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{a (64 a B-105 A c x) \left (a+c x^2\right )^{3/2}}{840 c^3}+\frac{\left (a^3 A\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c^2}\\ &=\frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{a (64 a B-105 A c x) \left (a+c x^2\right )^{3/2}}{840 c^3}+\frac{\left (a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{16 c^2}\\ &=\frac{a^2 A x \sqrt{a+c x^2}}{16 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac{A x^3 \left (a+c x^2\right )^{3/2}}{6 c}+\frac{B x^4 \left (a+c x^2\right )^{3/2}}{7 c}+\frac{a (64 a B-105 A c x) \left (a+c x^2\right )^{3/2}}{840 c^3}+\frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.212869, size = 113, normalized size = 0.74 \[ \frac{\sqrt{a+c x^2} \left (-a^2 c x (105 A+64 B x)+\frac{105 a^{5/2} A \sqrt{c} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}+128 a^3 B+2 a c^2 x^3 (35 A+24 B x)+40 c^3 x^5 (7 A+6 B x)\right )}{1680 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 136, normalized size = 0.9 \begin{align*}{\frac{B{x}^{4}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{4\,aB{x}^{2}}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{8\,B{a}^{2}}{105\,{c}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{x}^{3}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aAx}{8\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}Ax}{16\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{A{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \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 = 1.67463, size = 558, normalized size = 3.67 \begin{align*} \left [\frac{105 \, A a^{3} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 48 \, B a c^{2} x^{4} + 70 \, A a c^{2} x^{3} - 64 \, B a^{2} c x^{2} - 105 \, A a^{2} c x + 128 \, B a^{3}\right )} \sqrt{c x^{2} + a}}{3360 \, c^{3}}, -\frac{105 \, A a^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 48 \, B a c^{2} x^{4} + 70 \, A a c^{2} x^{3} - 64 \, B a^{2} c x^{2} - 105 \, A a^{2} c x + 128 \, B a^{3}\right )} \sqrt{c x^{2} + a}}{1680 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 8.35418, size = 216, normalized size = 1.42 \begin{align*} - \frac{A a^{\frac{5}{2}} x}{16 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{\frac{3}{2}} x^{3}}{48 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 A \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{5}{2}}} + \frac{A c x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15849, size = 143, normalized size = 0.94 \begin{align*} -\frac{A a^{3} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{5}{2}}} + \frac{1}{1680} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \, B x + 7 \, A\right )} x + \frac{6 \, B a}{c}\right )} x + \frac{35 \, A a}{c}\right )} x - \frac{32 \, B a^{2}}{c^{2}}\right )} x - \frac{105 \, A a^{2}}{c^{2}}\right )} x + \frac{128 \, B a^{3}}{c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]