Optimal. Leaf size=107 \[ \frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {5 a^3 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}} \]
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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {655, 201, 223,
212} \begin {gather*} \frac {5 a^3 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}}+\frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 655
Rubi steps
\begin {align*} \int (d+e x) \left (a+c x^2\right )^{5/2} \, dx &=\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+d \int \left (a+c x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{6} (5 a d) \int \left (a+c x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{8} \left (5 a^2 d\right ) \int \sqrt {a+c x^2} \, dx\\ &=\frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{16} \left (5 a^3 d\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {1}{16} \left (5 a^3 d\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {5}{16} a^2 d x \sqrt {a+c x^2}+\frac {5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac {1}{6} d x \left (a+c x^2\right )^{5/2}+\frac {e \left (a+c x^2\right )^{7/2}}{7 c}+\frac {5 a^3 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 107, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^2} \left (48 a^3 e+8 c^3 x^5 (7 d+6 e x)+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)\right )-105 a^3 \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{336 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 86, normalized size = 0.80
method | result | size |
default | \(\frac {e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}+d \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )\) | \(86\) |
risch | \(\frac {\left (48 e \,c^{3} x^{6}+56 c^{3} d \,x^{5}+144 c^{2} a e \,x^{4}+182 a \,c^{2} d \,x^{3}+144 x^{2} a^{2} c e +231 a^{2} x c d +48 a^{3} e \right ) \sqrt {c \,x^{2}+a}}{336 c}+\frac {5 a^{3} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 \sqrt {c}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 78, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d x + \frac {5 \, a^{3} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e}{7 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 220, normalized size = 2.06 \begin {gather*} \left [\frac {105 \, a^{3} \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (56 \, c^{3} d x^{5} + 182 \, a c^{2} d x^{3} + 231 \, a^{2} c d x + 48 \, {\left (c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}\right )} e\right )} \sqrt {c x^{2} + a}}{672 \, c}, -\frac {105 \, a^{3} \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (56 \, c^{3} d x^{5} + 182 \, a c^{2} d x^{3} + 231 \, a^{2} c d x + 48 \, {\left (c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}\right )} e\right )} \sqrt {c x^{2} + a}}{336 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.65, size = 348, normalized size = 3.25 \begin {gather*} \frac {a^{\frac {5}{2}} d x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} d x}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} c d x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 \sqrt {a} c^{2} d x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {5 a^{3} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 \sqrt {c}} + a^{2} e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + 2 a c e \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c^{2} e \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 ) + \frac {c^{3} d x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.31, size = 105, normalized size = 0.98 \begin {gather*} -\frac {5 \, a^{3} d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, \sqrt {c}} + \frac {1}{336} \, \sqrt {c x^{2} + a} {\left (\frac {48 \, a^{3} e}{c} + {\left (231 \, a^{2} d + 2 \, {\left (72 \, a^{2} e + {\left (91 \, a c d + 4 \, {\left (18 \, a c e + {\left (6 \, c^{2} x e + 7 \, c^{2} d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 54, normalized size = 0.50 \begin {gather*} \frac {e\,{\left (c\,x^2+a\right )}^{7/2}}{7\,c}+\frac {d\,x\,{\left (c\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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