Optimal. Leaf size=376 \[ -\frac {\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \]
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Rubi [A] time = 0.37, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac {\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b x} (c+d x)^{5/2} \, dx &=\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\int \sqrt {a+b x} (c+d x)^{5/2} \left (-a c-\frac {1}{2} (5 b c+9 a d) x\right ) \, dx}{6 b d}\\ &=-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \int \sqrt {a+b x} (c+d x)^{5/2} \, dx}{40 b^2 d^2}\\ &=\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{64 b^3 d^2}\\ &=\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{128 b^4 d^2}\\ &=\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^5 d^2}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^5 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 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}\right )}{512 b^6 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 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}}{\sqrt {c+d x}}\right )}{512 b^6 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {(b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 2.01, size = 320, normalized size = 0.85 \[ \frac {(a+b x)^{3/2} (c+d x)^{7/2} \left (\frac {3 \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 b^5 d^2 (a+b x)^2 (b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (15 a^2 d^2-10 a b d (5 c+2 d x)+b^2 \left (59 c^2+68 c d x+24 d^2 x^2\right )\right )+15 b^5 d (a+b x) (b c-a d)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}}-15 b^5 \sqrt {d} \sqrt {a+b x} (b c-a d)^5 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{128 b^9 d^2 (a+b x)^2 (c+d x)^4 \sqrt {b c-a d}}-3 (9 a d+5 b c)+30 b d x\right )}{180 b^2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 892, normalized size = 2.37 \[ \left [\frac {15 \, {\left (5 \, b^{6} c^{6} - 6 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 75 \, a^{4} b^{2} c^{2} d^{4} - 70 \, a^{5} b c d^{5} + 21 \, a^{6} d^{6}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (1280 \, b^{6} d^{6} x^{5} + 75 \, b^{6} c^{5} d - 65 \, a b^{5} c^{4} d^{2} - 90 \, a^{2} b^{4} c^{3} d^{3} + 838 \, a^{3} b^{3} c^{2} d^{4} - 945 \, a^{4} b^{2} c d^{5} + 315 \, a^{5} b d^{6} + 128 \, {\left (25 \, b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (135 \, b^{6} c^{2} d^{4} + 26 \, a b^{5} c d^{5} - 9 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (5 \, b^{6} c^{3} d^{3} + 51 \, a b^{5} c^{2} d^{4} - 61 \, a^{2} b^{4} c d^{5} + 21 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (25 \, b^{6} c^{4} d^{2} - 20 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 308 \, a^{3} b^{3} c d^{5} + 105 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{6} d^{4}}, \frac {15 \, {\left (5 \, b^{6} c^{6} - 6 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 75 \, a^{4} b^{2} c^{2} d^{4} - 70 \, a^{5} b c d^{5} + 21 \, a^{6} d^{6}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (1280 \, b^{6} d^{6} x^{5} + 75 \, b^{6} c^{5} d - 65 \, a b^{5} c^{4} d^{2} - 90 \, a^{2} b^{4} c^{3} d^{3} + 838 \, a^{3} b^{3} c^{2} d^{4} - 945 \, a^{4} b^{2} c d^{5} + 315 \, a^{5} b d^{6} + 128 \, {\left (25 \, b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (135 \, b^{6} c^{2} d^{4} + 26 \, a b^{5} c d^{5} - 9 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (5 \, b^{6} c^{3} d^{3} + 51 \, a b^{5} c^{2} d^{4} - 61 \, a^{2} b^{4} c d^{5} + 21 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (25 \, b^{6} c^{4} d^{2} - 20 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 308 \, a^{3} b^{3} c d^{5} + 105 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{6} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.05, size = 2043, normalized size = 5.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1240, normalized size = 3.30 \[ -\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (-2560 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} d^{5} x^{5}-256 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} d^{5} x^{4}-6400 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c \,d^{4} x^{4}+315 a^{6} d^{6} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-1050 a^{5} b c \,d^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+1125 a^{4} b^{2} c^{2} d^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-300 a^{3} b^{3} c^{3} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-75 a^{2} b^{4} c^{4} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+288 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} d^{5} x^{3}-90 a \,b^{5} c^{5} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-832 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c \,d^{4} x^{3}+75 b^{6} c^{6} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-4320 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{2} d^{3} x^{3}-336 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} d^{5} x^{2}+976 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c \,d^{4} x^{2}-816 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{2} d^{3} x^{2}-80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{3} d^{2} x^{2}+420 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,d^{5} x -1232 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c \,d^{4} x +1048 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{2} d^{3} x -80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{3} d^{2} x +100 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{4} d x -630 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} d^{5}+1890 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b c \,d^{4}-1676 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{2} d^{3}+180 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{3} d^{2}+130 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{4} d -150 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{5}\right )}{15360 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{5} d^{3}} \]
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 \[ \int x^2\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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