Optimal. Leaf size=149 \[ \frac {5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}+\frac {5 a^2 x \sqrt {a+b x^2} (8 b c-a d)}{128 b}+\frac {x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac {5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b} \]
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Rubi [A] time = 0.05, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {388, 195, 217, 206} \[ \frac {5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}+\frac {5 a^2 x \sqrt {a+b x^2} (8 b c-a d)}{128 b}+\frac {x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac {5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 388
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \, dx &=\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {(-8 b c+a d) \int \left (a+b x^2\right )^{5/2} \, dx}{8 b}\\ &=\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {(5 a (8 b c-a d)) \int \left (a+b x^2\right )^{3/2} \, dx}{48 b}\\ &=\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {\left (5 a^2 (8 b c-a d)\right ) \int \sqrt {a+b x^2} \, dx}{64 b}\\ &=\frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {\left (5 a^3 (8 b c-a d)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b}\\ &=\frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {\left (5 a^3 (8 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b}\\ &=\frac {5 a^2 (8 b c-a d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 b c-a d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 b c-a d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {d x \left (a+b x^2\right )^{7/2}}{8 b}+\frac {5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 130, normalized size = 0.87 \[ \frac {\sqrt {a+b x^2} \left (\sqrt {b} x \left (15 a^3 d+2 a^2 b \left (132 c+59 d x^2\right )+8 a b^2 x^2 \left (26 c+17 d x^2\right )+16 b^3 x^4 \left (4 c+3 d x^2\right )\right )-\frac {15 a^{5/2} (a d-8 b c) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{384 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 260, normalized size = 1.74 \[ \left [-\frac {15 \, {\left (8 \, a^{3} b c - a^{4} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, b^{4} d x^{7} + 8 \, {\left (8 \, b^{4} c + 17 \, a b^{3} d\right )} x^{5} + 2 \, {\left (104 \, a b^{3} c + 59 \, a^{2} b^{2} d\right )} x^{3} + 3 \, {\left (88 \, a^{2} b^{2} c + 5 \, a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{2}}, -\frac {15 \, {\left (8 \, a^{3} b c - a^{4} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d x^{7} + 8 \, {\left (8 \, b^{4} c + 17 \, a b^{3} d\right )} x^{5} + 2 \, {\left (104 \, a b^{3} c + 59 \, a^{2} b^{2} d\right )} x^{3} + 3 \, {\left (88 \, a^{2} b^{2} c + 5 \, a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 135, normalized size = 0.91 \[ \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b^{2} d x^{2} + \frac {8 \, b^{8} c + 17 \, a b^{7} d}{b^{6}}\right )} x^{2} + \frac {104 \, a b^{7} c + 59 \, a^{2} b^{6} d}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (88 \, a^{2} b^{6} c + 5 \, a^{3} b^{5} d\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {5 \, {\left (8 \, a^{3} b c - a^{4} d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 166, normalized size = 1.11 \[ -\frac {5 a^{4} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}+\frac {5 a^{3} c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 \sqrt {b}}-\frac {5 \sqrt {b \,x^{2}+a}\, a^{3} d x}{128 b}+\frac {5 \sqrt {b \,x^{2}+a}\, a^{2} c x}{16}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} d x}{192 b}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a c x}{24}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a d x}{48 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} c x}{6}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} d x}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 151, normalized size = 1.01 \[ \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c x + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} d x}{128 \, b} + \frac {5 \, a^{3} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, a^{4} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} \]
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 \[ \int {\left (b\,x^2+a\right )}^{5/2}\,\left (d\,x^2+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 29.53, size = 316, normalized size = 2.12 \[ \frac {5 a^{\frac {7}{2}} d x}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {5}{2}} c x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} c x}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} d x^{3}}{384 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} b c x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} b d x^{5}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 \sqrt {a} b^{2} c x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{2} d x^{7}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 a^{4} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {3}{2}}} + \frac {5 a^{3} c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {b^{3} c x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} d x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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