Optimal. Leaf size=152 \[ \frac {a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{5/2}}+\frac {a^2 A x \sqrt {a+c x^2}}{16 c^2}+\frac {a \left (a+c x^2\right )^{3/2} (64 a B-105 A c x)}{840 c^3}+\frac {A x^3 \left (a+c x^2\right )^{3/2}}{6 c}-\frac {4 a B x^2 \left (a+c x^2\right )^{3/2}}{35 c^2}+\frac {B x^4 \left (a+c x^2\right )^{3/2}}{7 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {833, 780, 195, 217, 206} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 780
Rule 833
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 = 0.21, size = 113, normalized size = 0.74 \begin {gather*} \frac {\sqrt {a+c x^2} \left (\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-a^2 c x (105 A+64 B x)+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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.31, size = 116, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a+c x^2} \left (128 a^3 B-105 a^2 A c x-64 a^2 B c x^2+70 a A c^2 x^3+48 a B c^2 x^4+280 A c^3 x^5+240 B c^3 x^6\right )}{1680 c^3}-\frac {a^3 A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{16 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 224, normalized size = 1.47 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 106, normalized size = 0.70 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 136, normalized size = 0.89 \begin {gather*} \frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,x^{4}}{7 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,x^{3}}{6 c}+\frac {A \,a^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{2}+a}\, A \,a^{2} x}{16 c^{2}}-\frac {4 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B a \,x^{2}}{35 c^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A a x}{8 c^{2}}+\frac {8 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{2}}{105 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 128, normalized size = 0.84 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B x^{4}}{7 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A x^{3}}{6 \, c} - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B a x^{2}}{35 \, c^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A a x}{8 \, c^{2}} + \frac {\sqrt {c x^{2} + a} A a^{2} x}{16 \, c^{2}} + \frac {A a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {5}{2}}} + \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B a^{2}}{105 \, c^{3}} \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 x^4\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 8.36, size = 216, normalized size = 1.42 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________