Optimal. Leaf size=376 \[ \frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{11/2}} \]
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Rubi [A]
time = 0.25, 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 = {92, 81, 52, 65,
223, 212} \begin {gather*} -\frac {\left (5 a^2 d^2+14 a b c d+21 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^{7/2} d^{11/2}}+\frac {(a+b x)^{7/2} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right )}{160 b^3 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^3}{512 b^3 d^5}-\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^2}{768 b^3 d^4}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)}{960 b^3 d^3}-\frac {(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+9 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rubi steps
\begin {align*} \int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx &=\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\int (a+b x)^{5/2} \sqrt {c+d x} \left (-a c-\frac {1}{2} (9 b c+5 a d) x\right ) \, dx}{6 b d}\\ &=-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \int (a+b x)^{5/2} \sqrt {c+d x} \, dx}{40 b^2 d^2}\\ &=\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\left ((b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{320 b^3 d^2}\\ &=\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{384 b^3 d^3}\\ &=-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\left ((b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^3 d^4}\\ &=\frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^3 d^5}\\ &=\frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \text {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^4 d^5}\\ &=\frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \text {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^4 d^5}\\ &=\frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 306, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (13 c+10 d x)+10 a^3 b^2 d^3 \left (-9 c^2+4 c d x+4 d^2 x^2\right )+2 a^2 b^3 d^2 \left (419 c^3-262 c^2 d x+204 c d^2 x^2+1080 d^3 x^3\right )+a b^4 d \left (-945 c^4+616 c^3 d x-488 c^2 d^2 x^2+416 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (315 c^5-210 c^4 d x+168 c^3 d^2 x^2-144 c^2 d^3 x^3+128 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^5}-\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{512 b^{7/2} d^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs.
\(2(326)=652\).
time = 0.07, size = 1037, normalized size = 2.76
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-4320 a^{2} b^{3} d^{5} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+288 b^{5} c^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-80 a^{3} b^{2} d^{5} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-336 b^{5} c^{3} d^{2} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-6400 a \,b^{4} d^{5} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-256 b^{5} c \,d^{4} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-150 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} d^{5}-630 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{5}+100 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,d^{5} x +420 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{4} d x +130 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b c \,d^{4}+180 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{2} d^{3}-1676 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{3} d^{2}+1890 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{4} d -80 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c \,d^{4} x +1048 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{2} d^{3} x -1232 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{3} d^{2} x +75 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{6} d^{6}+315 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{6} c^{6}-90 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} b c \,d^{5}-2560 b^{5} d^{5} x^{5} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-75 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b^{2} c^{2} d^{4}-300 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{3} c^{3} d^{3}+1125 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{4} c^{4} d^{2}-1050 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{5} c^{5} d -832 a \,b^{4} c \,d^{4} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-816 a^{2} b^{3} c \,d^{4} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+976 a \,b^{4} c^{2} d^{3} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\right )}{15360 b^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, d^{5} \sqrt {b d}}\) | \(1037\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 892, normalized size = 2.37 \begin {gather*} \left [\frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, 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} + 315 \, b^{6} c^{5} d - 945 \, a b^{5} c^{4} d^{2} + 838 \, a^{2} b^{4} c^{3} d^{3} - 90 \, a^{3} b^{3} c^{2} d^{4} - 65 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{6} c^{2} d^{4} - 26 \, a b^{5} c d^{5} - 135 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (21 \, b^{6} c^{3} d^{3} - 61 \, a b^{5} c^{2} d^{4} + 51 \, a^{2} b^{4} c d^{5} + 5 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (105 \, b^{6} c^{4} d^{2} - 308 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 20 \, a^{3} b^{3} c d^{5} + 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{4} d^{6}}, \frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, 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} + 315 \, b^{6} c^{5} d - 945 \, a b^{5} c^{4} d^{2} + 838 \, a^{2} b^{4} c^{3} d^{3} - 90 \, a^{3} b^{3} c^{2} d^{4} - 65 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{6} c^{2} d^{4} - 26 \, a b^{5} c d^{5} - 135 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (21 \, b^{6} c^{3} d^{3} - 61 \, a b^{5} c^{2} d^{4} + 51 \, a^{2} b^{4} c d^{5} + 5 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (105 \, b^{6} c^{4} d^{2} - 308 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 20 \, a^{3} b^{3} c d^{5} + 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{4} d^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1361 vs.
\(2 (326) = 652\).
time = 1.32, size = 1361, normalized size = 3.62 \begin {gather*} \frac {12 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{4}} + \frac {b^{20} c d^{7} - 41 \, a b^{19} d^{8}}{b^{23} d^{8}}\right )} - \frac {7 \, b^{21} c^{2} d^{6} + 26 \, a b^{20} c d^{7} - 513 \, a^{2} b^{19} d^{8}}{b^{23} d^{8}}\right )} + \frac {5 \, {\left (7 \, b^{22} c^{3} d^{5} + 19 \, a b^{21} c^{2} d^{6} + 37 \, a^{2} b^{20} c d^{7} - 447 \, a^{3} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, b^{23} c^{4} d^{4} + 12 \, a b^{22} c^{3} d^{5} + 18 \, a^{2} b^{21} c^{2} d^{6} + 28 \, a^{3} b^{20} c d^{7} - 193 \, a^{4} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 10 \, a^{3} b^{2} c^{2} d^{3} + 35 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{3} d^{4}}\right )} a {\left | b \right |} + \frac {320 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} a^{3} {\left | b \right |}}{b^{2}} + \frac {120 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} a^{2} {\left | b \right |}}{b} + {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (2 \, {\left (b x + a\right )} {\left (8 \, {\left (b x + a\right )} {\left (\frac {10 \, {\left (b x + a\right )}}{b^{5}} + \frac {b^{30} c d^{9} - 61 \, a b^{29} d^{10}}{b^{34} d^{10}}\right )} - \frac {3 \, {\left (3 \, b^{31} c^{2} d^{8} + 14 \, a b^{30} c d^{9} - 417 \, a^{2} b^{29} d^{10}\right )}}{b^{34} d^{10}}\right )} + \frac {21 \, b^{32} c^{3} d^{7} + 77 \, a b^{31} c^{2} d^{8} + 183 \, a^{2} b^{30} c d^{9} - 3481 \, a^{3} b^{29} d^{10}}{b^{34} d^{10}}\right )} {\left (b x + a\right )} - \frac {5 \, {\left (21 \, b^{33} c^{4} d^{6} + 56 \, a b^{32} c^{3} d^{7} + 106 \, a^{2} b^{31} c^{2} d^{8} + 176 \, a^{3} b^{30} c d^{9} - 2279 \, a^{4} b^{29} d^{10}\right )}}{b^{34} d^{10}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (21 \, b^{34} c^{5} d^{5} + 35 \, a b^{33} c^{4} d^{6} + 50 \, a^{2} b^{32} c^{3} d^{7} + 70 \, a^{3} b^{31} c^{2} d^{8} + 105 \, a^{4} b^{30} c d^{9} - 793 \, a^{5} b^{29} d^{10}\right )}}{b^{34} d^{10}}\right )} \sqrt {b x + a} + \frac {15 \, {\left (21 \, b^{6} c^{6} + 14 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} + 20 \, a^{3} b^{3} c^{3} d^{3} + 35 \, a^{4} b^{2} c^{2} d^{4} + 126 \, a^{5} b c d^{5} - 231 \, a^{6} d^{6}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{4} d^{5}}\right )} b {\left | b \right |}}{7680 \, b} \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 = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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