Optimal. Leaf size=110 \[ \frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}}+\frac {e \sqrt {a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac {e \sqrt {a+c x^2} (d+e x)^2}{3 c} \]
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Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {743, 780, 217, 206} \begin {gather*} \frac {e \sqrt {a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}}+\frac {e \sqrt {a+c x^2} (d+e x)^2}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 743
Rule 780
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\sqrt {a+c x^2}} \, dx &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {\int \frac {(d+e x) \left (3 c d^2-2 a e^2+5 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{3 c}\\ &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {\left (d \left (2 c d^2-3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c}\\ &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {\left (d \left (2 c d^2-3 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 )}{2 c}\\ &=\frac {e (d+e x)^2 \sqrt {a+c x^2}}{3 c}+\frac {e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt {a+c x^2}}{6 c^2}+\frac {d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 92, normalized size = 0.84 \begin {gather*} \frac {e \sqrt {a+c x^2} \left (c \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 a e^2\right )+3 \sqrt {c} d \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 94, normalized size = 0.85 \begin {gather*} \frac {\left (3 a d e^2-2 c d^3\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 c^{3/2}}+\frac {\sqrt {a+c x^2} \left (-4 a e^3+18 c d^2 e+9 c d e^2 x+2 c e^3 x^2\right )}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 180, normalized size = 1.64 \begin {gather*} \left [-\frac {3 \, {\left (2 \, c d^{3} - 3 \, a 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 (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, c^{2}}, -\frac {3 \, {\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 90, normalized size = 0.82 \begin {gather*} \frac {1}{6} \, \sqrt {c x^{2} + a} {\left (x {\left (\frac {2 \, x e^{3}}{c} + \frac {9 \, d e^{2}}{c}\right )} + \frac {2 \, {\left (9 \, c^{2} d^{2} e - 2 \, a c e^{3}\right )}}{c^{3}}\right )} - \frac {{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 126, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c \,x^{2}+a}\, e^{3} x^{2}}{3 c}-\frac {3 a d \,e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}+\frac {d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}+\frac {3 \sqrt {c \,x^{2}+a}\, d \,e^{2} x}{2 c}-\frac {2 \sqrt {c \,x^{2}+a}\, a \,e^{3}}{3 c^{2}}+\frac {3 \sqrt {c \,x^{2}+a}\, d^{2} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 111, normalized size = 1.01 \begin {gather*} \frac {\sqrt {c x^{2} + a} e^{3} x^{2}}{3 \, c} + \frac {3 \, \sqrt {c x^{2} + a} d e^{2} x}{2 \, c} + \frac {d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} - \frac {3 \, a d e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {3}{2}}} + \frac {3 \, \sqrt {c x^{2} + a} d^{2} e}{c} - \frac {2 \, \sqrt {c x^{2} + a} a e^{3}}{3 \, c^{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 \frac {{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.86, size = 216, normalized size = 1.96 \begin {gather*} \frac {3 \sqrt {a} d e^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2 c} - \frac {3 a d e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 c^{\frac {3}{2}}} + d^{3} \left (\begin {cases} \frac {\sqrt {- \frac {a}{c}} \operatorname {asin}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c < 0 \\\frac {\sqrt {\frac {a}{c}} \operatorname {asinh}{\left (x \sqrt {\frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c > 0 \\\frac {\sqrt {- \frac {a}{c}} \operatorname {acosh}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {- a}} & \text {for}\: c > 0 \wedge a < 0 \end {cases}\right ) + 3 d^{2} e \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: c = 0 \\\frac {\sqrt {a + c x^{2}}}{c} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} - \frac {2 a \sqrt {a + c x^{2}}}{3 c^{2}} + \frac {x^{2} \sqrt {a + c x^{2}}}{3 c} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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