Optimal. Leaf size=340 \[ \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{256 b^2 d^5}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{384 b^2 d^4}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{480 b^2 d^3}-\frac {(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}}-\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d} \]
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Rubi [A] time = 0.42, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {446, 90, 80, 50, 63, 217, 206} \[ \frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{480 b^2 d^3}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{384 b^2 d^4}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{256 b^2 d^5}-\frac {(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}}-\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} \left (-a c-\frac {3}{2} (3 b c+a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^2\right )}{10 b d}\\ &=-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{160 b^2 d^2}\\ &=\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{192 b^2 d^3}\\ &=-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}+\frac {\left ((b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{256 b^2 d^4}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{512 b^2 d^5}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{256 b^3 d^5}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{256 b^3 d^5}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 271, normalized size = 0.80 \[ \frac {\sqrt {c+d x^2} \left (\frac {5 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {16 d^3 \left (a+b x^2\right )^3}{15 (b c-a d)^3}-\frac {4 d^2 \left (a+b x^2\right )^2}{3 (b c-a d)^2}+\frac {2 d \left (a+b x^2\right )}{b c-a d}-\frac {2 \sqrt {d} \sqrt {a+b x^2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}}\right )}{4 b d^5}-\frac {24 \left (a+b x^2\right )^4 (a d+3 b c)}{b d}+64 x^2 \left (a+b x^2\right )^4\right )}{640 b d \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 734, normalized size = 2.16 \[ \left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{8} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{6} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{4} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15360 \, b^{3} d^{6}}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (384 \, b^{5} d^{5} x^{8} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{6} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{4} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{7680 \, b^{3} d^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 398, normalized size = 1.17 \[ \frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (6 \, {\left (b x^{2} + a\right )} {\left (\frac {8 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac {63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac {5 \, {\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} + \frac {15 \, {\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} + \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{5}}\right )} b}{3840 \, {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1054, normalized size = 3.10 \[ \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (768 \sqrt {b d}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, b^{4} d^{4} x^{8}+2016 \sqrt {b d}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, a \,b^{3} d^{4} x^{6}-864 \sqrt {b d}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, b^{4} c \,d^{3} x^{6}+1488 \sqrt {b d}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} b^{2} d^{4} x^{4}-2368 \sqrt {b d}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, a \,b^{3} c \,d^{3} x^{4}+1008 \sqrt {b d}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, b^{4} c^{2} d^{2} x^{4}+45 a^{5} d^{5} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+75 a^{4} b c \,d^{4} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+450 a^{3} b^{2} c^{2} d^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-2250 a^{2} b^{3} c^{3} d^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2625 a \,b^{4} c^{4} d \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-945 b^{5} c^{5} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+60 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{3} b \,d^{4} x^{2}-1924 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x^{2}+2996 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x^{2}-1260 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{4} c^{3} d \,x^{2}-90 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{4} d^{4}-180 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{3} b c \,d^{3}+3128 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-4620 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{3} c^{3} d +1890 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{4} c^{4}\right )}{7680 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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