Optimal. Leaf size=114 \[ -\frac {b \left (c+d x^2\right )^{7/2} (3 b c-2 a d)}{7 d^4}+\frac {\left (c+d x^2\right )^{5/2} (b c-a d) (3 b c-a d)}{5 d^4}-\frac {c \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^4}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^4} \]
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Rubi [A] time = 0.10, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {446, 77} \begin {gather*} -\frac {b \left (c+d x^2\right )^{7/2} (3 b c-2 a d)}{7 d^4}+\frac {\left (c+d x^2\right )^{5/2} (b c-a d) (3 b c-a d)}{5 d^4}-\frac {c \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^4}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^2 \sqrt {c+d x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {c (b c-a d)^2 \sqrt {c+d x}}{d^3}+\frac {(b c-a d) (3 b c-a d) (c+d x)^{3/2}}{d^3}-\frac {b (3 b c-2 a d) (c+d x)^{5/2}}{d^3}+\frac {b^2 (c+d x)^{7/2}}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}{3 d^4}+\frac {(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{5/2}}{5 d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{7/2}}{7 d^4}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 99, normalized size = 0.87 \begin {gather*} \frac {\left (c+d x^2\right )^{3/2} \left (21 a^2 d^2 \left (3 d x^2-2 c\right )+6 a b d \left (8 c^2-12 c d x^2+15 d^2 x^4\right )+b^2 \left (-16 c^3+24 c^2 d x^2-30 c d^2 x^4+35 d^3 x^6\right )\right )}{315 d^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 111, normalized size = 0.97 \begin {gather*} \frac {\left (c+d x^2\right )^{3/2} \left (-42 a^2 c d^2+63 a^2 d^3 x^2+48 a b c^2 d-72 a b c d^2 x^2+90 a b d^3 x^4-16 b^2 c^3+24 b^2 c^2 d x^2-30 b^2 c d^2 x^4+35 b^2 d^3 x^6\right )}{315 d^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 140, normalized size = 1.23 \begin {gather*} \frac {{\left (35 \, b^{2} d^{4} x^{8} + 5 \, {\left (b^{2} c d^{3} + 18 \, a b d^{4}\right )} x^{6} - 16 \, b^{2} c^{4} + 48 \, a b c^{3} d - 42 \, a^{2} c^{2} d^{2} - 3 \, {\left (2 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - 21 \, a^{2} d^{4}\right )} x^{4} + {\left (8 \, b^{2} c^{3} d - 24 \, a b c^{2} d^{2} + 21 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{315 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 150, normalized size = 1.32 \begin {gather*} \frac {35 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} - 135 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c + 189 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} - 105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} + 90 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d - 252 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d + 210 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d + 63 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2} - 105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{2}}{315 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 108, normalized size = 0.95 \begin {gather*} -\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (-35 b^{2} x^{6} d^{3}-90 a b \,d^{3} x^{4}+30 b^{2} c \,d^{2} x^{4}-63 a^{2} d^{3} x^{2}+72 a b c \,d^{2} x^{2}-24 b^{2} c^{2} d \,x^{2}+42 a^{2} c \,d^{2}-48 a b \,c^{2} d +16 b^{2} c^{3}\right )}{315 d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 181, normalized size = 1.59 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} x^{6}}{9 \, d} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x^{4}}{21 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b x^{4}}{7 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} x^{2}}{105 \, d^{3}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c x^{2}}{35 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} x^{2}}{5 \, d} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3}}{315 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2}}{105 \, d^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c}{15 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 137, normalized size = 1.20 \begin {gather*} \sqrt {d\,x^2+c}\,\left (\frac {b^2\,x^8}{9}-\frac {42\,a^2\,c^2\,d^2-48\,a\,b\,c^3\,d+16\,b^2\,c^4}{315\,d^4}+\frac {x^4\,\left (63\,a^2\,d^4+18\,a\,b\,c\,d^3-6\,b^2\,c^2\,d^2\right )}{315\,d^4}+\frac {b\,x^6\,\left (18\,a\,d+b\,c\right )}{63\,d}+\frac {c\,x^2\,\left (21\,a^2\,d^2-24\,a\,b\,c\,d+8\,b^2\,c^2\right )}{315\,d^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.24, size = 308, normalized size = 2.70 \begin {gather*} \begin {cases} - \frac {2 a^{2} c^{2} \sqrt {c + d x^{2}}}{15 d^{2}} + \frac {a^{2} c x^{2} \sqrt {c + d x^{2}}}{15 d} + \frac {a^{2} x^{4} \sqrt {c + d x^{2}}}{5} + \frac {16 a b c^{3} \sqrt {c + d x^{2}}}{105 d^{3}} - \frac {8 a b c^{2} x^{2} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {2 a b c x^{4} \sqrt {c + d x^{2}}}{35 d} + \frac {2 a b x^{6} \sqrt {c + d x^{2}}}{7} - \frac {16 b^{2} c^{4} \sqrt {c + d x^{2}}}{315 d^{4}} + \frac {8 b^{2} c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {2 b^{2} c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {b^{2} c x^{6} \sqrt {c + d x^{2}}}{63 d} + \frac {b^{2} x^{8} \sqrt {c + d x^{2}}}{9} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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