Optimal. Leaf size=161 \[ \frac {\left (3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}+\frac {e \sqrt {a+c x^2} \left (e x \left (26 c d^2-9 a e^2\right )+4 d \left (19 c d^2-16 a e^2\right )\right )}{24 c^2}+\frac {e \sqrt {a+c x^2} (d+e x)^3}{4 c}+\frac {7 d e \sqrt {a+c x^2} (d+e x)^2}{12 c} \]
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Rubi [A] time = 0.16, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {743, 833, 780, 217, 206} \begin {gather*} \frac {\left (3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}+\frac {e \sqrt {a+c x^2} \left (e x \left (26 c d^2-9 a e^2\right )+4 d \left (19 c d^2-16 a e^2\right )\right )}{24 c^2}+\frac {e \sqrt {a+c x^2} (d+e x)^3}{4 c}+\frac {7 d e \sqrt {a+c x^2} (d+e x)^2}{12 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 743
Rule 780
Rule 833
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\sqrt {a+c x^2}} \, dx &=\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {\int \frac {(d+e x)^2 \left (4 c d^2-3 a e^2+7 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{4 c}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {\int \frac {(d+e x) \left (c d \left (12 c d^2-23 a e^2\right )+c e \left (26 c d^2-9 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{12 c^2}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c^2}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c^2}\\ &=\frac {7 d e (d+e x)^2 \sqrt {a+c x^2}}{12 c}+\frac {e (d+e x)^3 \sqrt {a+c x^2}}{4 c}+\frac {e \left (4 d \left (19 c d^2-16 a e^2\right )+e \left (26 c d^2-9 a e^2\right ) x\right ) \sqrt {a+c x^2}}{24 c^2}+\frac {\left (8 c^2 d^4-24 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 126, normalized size = 0.78 \begin {gather*} \frac {3 \left (3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )+\sqrt {c} e \sqrt {a+c x^2} \left (c \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )-a e^2 (64 d+9 e x)\right )}{24 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 127, normalized size = 0.79 \begin {gather*} \frac {\left (-3 a^2 e^4+24 a c d^2 e^2-8 c^2 d^4\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 c^{5/2}}+\frac {\sqrt {a+c x^2} \left (-64 a d e^3-9 a e^4 x+96 c d^3 e+72 c d^2 e^2 x+32 c d e^3 x^2+6 c e^4 x^3\right )}{24 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 270, normalized size = 1.68 \begin {gather*} \left [\frac {3 \, {\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, c^{2} e^{4} x^{3} + 32 \, c^{2} d e^{3} x^{2} + 96 \, c^{2} d^{3} e - 64 \, a c d e^{3} + 9 \, {\left (8 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{48 \, c^{3}}, -\frac {3 \, {\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (6 \, c^{2} e^{4} x^{3} + 32 \, c^{2} d e^{3} x^{2} + 96 \, c^{2} d^{3} e - 64 \, a c d e^{3} + 9 \, {\left (8 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{24 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 133, normalized size = 0.83 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, x {\left (\frac {3 \, x e^{4}}{c} + \frac {16 \, d e^{3}}{c}\right )} + \frac {9 \, {\left (8 \, c^{3} d^{2} e^{2} - a c^{2} e^{4}\right )}}{c^{4}}\right )} x + \frac {32 \, {\left (3 \, c^{3} d^{3} e - 2 \, a c^{2} d e^{3}\right )}}{c^{4}}\right )} - \frac {{\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 198, normalized size = 1.23 \begin {gather*} \frac {\sqrt {c \,x^{2}+a}\, e^{4} x^{3}}{4 c}+\frac {4 \sqrt {c \,x^{2}+a}\, d \,e^{3} x^{2}}{3 c}+\frac {3 a^{2} e^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {5}{2}}}-\frac {3 a \,d^{2} e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}+\frac {d^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}-\frac {3 \sqrt {c \,x^{2}+a}\, a \,e^{4} x}{8 c^{2}}+\frac {3 \sqrt {c \,x^{2}+a}\, d^{2} e^{2} x}{c}-\frac {8 \sqrt {c \,x^{2}+a}\, a d \,e^{3}}{3 c^{2}}+\frac {4 \sqrt {c \,x^{2}+a}\, d^{3} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 176, normalized size = 1.09 \begin {gather*} \frac {\sqrt {c x^{2} + a} e^{4} x^{3}}{4 \, c} + \frac {4 \, \sqrt {c x^{2} + a} d e^{3} x^{2}}{3 \, c} + \frac {3 \, \sqrt {c x^{2} + a} d^{2} e^{2} x}{c} - \frac {3 \, \sqrt {c x^{2} + a} a e^{4} x}{8 \, c^{2}} + \frac {d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} - \frac {3 \, a d^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}}} + \frac {3 \, a^{2} e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {5}{2}}} + \frac {4 \, \sqrt {c x^{2} + a} d^{3} e}{c} - \frac {8 \, \sqrt {c x^{2} + a} a d 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 )}^4}{\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 = 8.67, size = 330, normalized size = 2.05 \begin {gather*} - \frac {3 a^{\frac {3}{2}} e^{4} x}{8 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} d^{2} e^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{c} - \frac {\sqrt {a} e^{4} x^{3}}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{2} e^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {5}{2}}} - \frac {3 a d^{2} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{c^{\frac {3}{2}}} + d^{4} \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 ) + 4 d^{3} 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 ) + 4 d 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 ) + \frac {e^{4} x^{5}}{4 \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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