Optimal. Leaf size=169 \[ \frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {(2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {427, 542, 396,
223, 212} \begin {gather*} \frac {(2 b c-a d) \left (5 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{7/2}}+\frac {d x \sqrt {a+b x^2} \left (15 a^2 d^2-44 a b c d+44 b^2 c^2\right )}{48 b^3}+\frac {5 d x \sqrt {a+b x^2} \left (c+d x^2\right ) (2 b c-a d)}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 396
Rule 427
Rule 542
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\sqrt {a+b x^2}} \, dx &=\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (6 b c-a d)+5 d (2 b c-a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{6 b}\\ &=\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\int \frac {c \left (24 b^2 c^2-14 a b c d+5 a^2 d^2\right )+d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x^2}{\sqrt {a+b x^2}} \, dx}{24 b^2}\\ &=\frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\left ((2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b^3}\\ &=\frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {\left ((2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^3}\\ &=\frac {d \left (44 b^2 c^2-44 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2}}{48 b^3}+\frac {5 d (2 b c-a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 b^2}+\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 b}+\frac {(2 b c-a d) \left (8 b^2 c^2-8 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 138, normalized size = 0.82 \begin {gather*} \frac {\sqrt {b} d x \sqrt {a+b x^2} \left (15 a^2 d^2-2 a b d \left (27 c+5 d x^2\right )+4 b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+\left (-48 b^3 c^3+72 a b^2 c^2 d-54 a^2 b c d^2+15 a^3 d^3\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 227, normalized size = 1.34
method | result | size |
risch | \(\frac {d x \left (8 b^{2} d^{2} x^{4}-10 a b \,d^{2} x^{2}+36 b^{2} c d \,x^{2}+15 a^{2} d^{2}-54 a b c d +72 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{3}}-\frac {5 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{3} d^{3}}{16 b^{\frac {7}{2}}}+\frac {9 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{2} c \,d^{2}}{8 b^{\frac {5}{2}}}-\frac {3 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a \,c^{2} d}{2 b^{\frac {3}{2}}}+\frac {c^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(176\) |
default | \(d^{3} \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+3 c \,d^{2} \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+3 c^{2} d \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\frac {c^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 199, normalized size = 1.18 \begin {gather*} \frac {\sqrt {b x^{2} + a} d^{3} x^{5}}{6 \, b} + \frac {3 \, \sqrt {b x^{2} + a} c d^{2} x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a d^{3} x^{3}}{24 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} c^{2} d x}{2 \, b} - \frac {9 \, \sqrt {b x^{2} + a} a c d^{2} x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} d^{3} x}{16 \, b^{3}} + \frac {c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {3 \, a c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {9 \, a^{2} c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 300, normalized size = 1.78 \begin {gather*} \left [-\frac {3 \, {\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, b^{3} d^{3} x^{5} + 2 \, {\left (18 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{3} + 3 \, {\left (24 \, b^{3} c^{2} d - 18 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{4}}, -\frac {3 \, {\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{3} d^{3} x^{5} + 2 \, {\left (18 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{3} + 3 \, {\left (24 \, b^{3} c^{2} d - 18 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.17, size = 400, normalized size = 2.37 \begin {gather*} \frac {5 a^{\frac {5}{2}} d^{3} x}{16 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {9 a^{\frac {3}{2}} c d^{2} x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {3}{2}} d^{3} x^{3}}{48 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 \sqrt {a} c^{2} d x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {3 \sqrt {a} c d^{2} x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} d^{3} x^{5}}{24 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 a^{3} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {7}{2}}} + \frac {9 a^{2} c d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {3 a c^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + c^{3} \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) + \frac {3 c d^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {d^{3} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 150, normalized size = 0.89 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, d^{3} x^{2}}{b} + \frac {18 \, b^{4} c d^{2} - 5 \, a b^{3} d^{3}}{b^{5}}\right )} x^{2} + \frac {3 \, {\left (24 \, b^{4} c^{2} d - 18 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )}}{b^{5}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {7}{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\,x^2+c\right )}^3}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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