Optimal. Leaf size=154 \[ \frac {a^2 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}+\frac {x \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right )}{24 c}+\frac {a x \sqrt {a+c x^2} \left (6 c d^2-a e^2\right )}{16 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e \left (a+c x^2\right )^{5/2} (d+e x)}{6 c} \]
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Rubi [A] time = 0.07, antiderivative size = 154, 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 = {743, 641, 195, 217, 206} \[ \frac {a^2 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}+\frac {x \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right )}{24 c}+\frac {a x \sqrt {a+c x^2} \left (6 c d^2-a e^2\right )}{16 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e \left (a+c x^2\right )^{5/2} (d+e x)}{6 c} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rule 743
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (6 c d^2-a e^2+7 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (6 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (a \left (6 c d^2-a e^2\right )\right ) \int \sqrt {a+c x^2} \, dx}{8 c}\\ &=\frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (a^2 \left (6 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c}\\ &=\frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {\left (a^2 \left (6 c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c}\\ &=\frac {a \left (6 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{16 c}+\frac {\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac {e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac {a^2 \left (6 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 = 0.09, size = 132, normalized size = 0.86 \[ \frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^2 e (32 d+5 e x)+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )-15 a^2 \left (a e^2-6 c d^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{240 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 296, normalized size = 1.92 \[ \left [-\frac {15 \, {\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (40 \, c^{3} e^{2} x^{5} + 96 \, c^{3} d e x^{4} + 192 \, a c^{2} d e x^{2} + 96 \, a^{2} c d e + 10 \, {\left (6 \, c^{3} d^{2} + 7 \, a c^{2} e^{2}\right )} x^{3} + 15 \, {\left (10 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{480 \, c^{2}}, -\frac {15 \, {\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (40 \, c^{3} e^{2} x^{5} + 96 \, c^{3} d e x^{4} + 192 \, a c^{2} d e x^{2} + 96 \, a^{2} c d e + 10 \, {\left (6 \, c^{3} d^{2} + 7 \, a c^{2} e^{2}\right )} x^{3} + 15 \, {\left (10 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 142, normalized size = 0.92 \[ \frac {1}{240} \, \sqrt {c x^{2} + a} {\left (\frac {96 \, a^{2} d e}{c} + {\left (2 \, {\left (96 \, a d e + {\left (4 \, {\left (5 \, c x e^{2} + 12 \, c d e\right )} x + \frac {5 \, {\left (6 \, c^{5} d^{2} + 7 \, a c^{4} e^{2}\right )}}{c^{4}}\right )} x\right )} x + \frac {15 \, {\left (10 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )}}{c^{4}}\right )} x\right )} - \frac {{\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 161, normalized size = 1.05 \[ -\frac {a^{3} e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 a^{2} d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}-\frac {\sqrt {c \,x^{2}+a}\, a^{2} e^{2} x}{16 c}+\frac {3 \sqrt {c \,x^{2}+a}\, a \,d^{2} x}{8}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,e^{2} x}{24 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d^{2} x}{4}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} e^{2} x}{6 c}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} d e}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 146, normalized size = 0.95 \[ \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d^{2} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e^{2} x}{6 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{2} x}{24 \, c} - \frac {\sqrt {c x^{2} + a} a^{2} e^{2} x}{16 \, c} + \frac {3 \, a^{2} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} - \frac {a^{3} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d e}{5 \, c} \]
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 \[ \int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.78, size = 372, normalized size = 2.42 \[ \frac {a^{\frac {5}{2}} e^{2} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {3}{2}} d^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} d^{2} x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} e^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} c d^{2} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c e^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {a^{3} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {3 a^{2} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + 2 a d e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + 2 c d e \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} d^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} e^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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