Optimal. Leaf size=180 \[ \frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {757, 794, 201,
223, 212} \begin {gather*} \frac {3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}+\frac {e \left (a+c x^2\right )^{5/2} \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right )}{70 c^2}+\frac {d x \left (a+c x^2\right )^{3/2} \left (2 c d^2-a e^2\right )}{8 c}+\frac {3 a d x \sqrt {a+c x^2} \left (2 c d^2-a e^2\right )}{16 c}+\frac {e \left (a+c x^2\right )^{5/2} (d+e x)^2}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 757
Rule 794
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (7 c d^2-2 a e^2+9 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (d \left (2 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{2 c}\\ &=\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (3 a d \left (2 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{8 c}\\ &=\frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (3 a^2 d \left (2 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c}\\ &=\frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {\left (3 a^2 d \left (2 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c}\\ &=\frac {3 a d \left (2 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac {e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac {3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 173, normalized size = 0.96 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-32 a^3 e^3+a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )+4 c^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+2 a c^2 x \left (175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right )\right )+105 a^2 \sqrt {c} d \left (-2 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{560 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 191, normalized size = 1.06
method | result | size |
default | \(e^{3} \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {3 d^{2} e \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}+d^{3} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )\) | \(191\) |
risch | \(-\frac {\left (-80 e^{3} c^{3} x^{6}-280 d \,e^{2} c^{3} x^{5}-128 e^{3} c^{2} a \,x^{4}-336 d^{2} e \,c^{3} x^{4}-490 a \,c^{2} d \,e^{2} x^{3}-140 c^{3} d^{3} x^{3}-16 a^{2} c \,e^{3} x^{2}-672 c^{2} d^{2} a e \,x^{2}-105 d \,e^{2} a^{2} c x -350 d^{3} c^{2} a x +32 e^{3} a^{3}-336 d^{2} e \,a^{2} c \right ) \sqrt {c \,x^{2}+a}}{560 c^{2}}-\frac {3 a^{3} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) e^{2}}{16 c^{\frac {3}{2}}}+\frac {3 a^{2} d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 185, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{3} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d^{3} x + \frac {3 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} x^{2} e^{3}}{7 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} d x e^{2}}{2 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a d x e^{2}}{8 \, c} - \frac {3 \, \sqrt {c x^{2} + a} a^{2} d x e^{2}}{16 \, c} - \frac {3 \, a^{3} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{2}}{16 \, c^{\frac {3}{2}}} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} e}{5 \, c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a e^{3}}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.41, size = 374, normalized size = 2.08 \begin {gather*} \left [\frac {105 \, {\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (140 \, c^{3} d^{3} x^{3} + 350 \, a c^{2} d^{3} x + 16 \, {\left (5 \, c^{3} x^{6} + 8 \, a c^{2} x^{4} + a^{2} c x^{2} - 2 \, a^{3}\right )} e^{3} + 35 \, {\left (8 \, c^{3} d x^{5} + 14 \, a c^{2} d x^{3} + 3 \, a^{2} c d x\right )} e^{2} + 336 \, {\left (c^{3} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e\right )} \sqrt {c x^{2} + a}}{1120 \, c^{2}}, -\frac {105 \, {\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (140 \, c^{3} d^{3} x^{3} + 350 \, a c^{2} d^{3} x + 16 \, {\left (5 \, c^{3} x^{6} + 8 \, a c^{2} x^{4} + a^{2} c x^{2} - 2 \, a^{3}\right )} e^{3} + 35 \, {\left (8 \, c^{3} d x^{5} + 14 \, a c^{2} d x^{3} + 3 \, a^{2} c d x\right )} e^{2} + 336 \, {\left (c^{3} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e\right )} \sqrt {c x^{2} + a}}{560 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 10.66, size = 551, normalized size = 3.06 \begin {gather*} \frac {3 a^{\frac {5}{2}} d e^{2} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {3}{2}} d^{3} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} d^{3} x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} d e^{2} x^{3}}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} c d^{3} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c d e^{2} x^{5}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {3 a^{3} d e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {3 a^{2} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + 3 a d^{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 ) + a e^{3} \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 ) + 3 c d^{2} 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 e^{3} \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^{2} d^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} d e^{2} x^{7}}{2 \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 = 0.73, size = 212, normalized size = 1.18 \begin {gather*} \frac {1}{560} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, c x e^{3} + 7 \, c d e^{2}\right )} x + \frac {2 \, {\left (21 \, c^{6} d^{2} e + 8 \, a c^{5} e^{3}\right )}}{c^{5}}\right )} x + \frac {35 \, {\left (2 \, c^{6} d^{3} + 7 \, a c^{5} d e^{2}\right )}}{c^{5}}\right )} x + \frac {8 \, {\left (42 \, a c^{5} d^{2} e + a^{2} c^{4} e^{3}\right )}}{c^{5}}\right )} x + \frac {35 \, {\left (10 \, a c^{5} d^{3} + 3 \, a^{2} c^{4} d e^{2}\right )}}{c^{5}}\right )} x + \frac {16 \, {\left (21 \, a^{2} c^{4} d^{2} e - 2 \, a^{3} c^{3} e^{3}\right )}}{c^{5}}\right )} - \frac {3 \, {\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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