Optimal. Leaf size=349 \[ \frac {(b c-a d)^3 \left (4 a b c d-7 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^4 d^4}-\frac {(b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^4 d^3}-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {(b c-a d)^4 \left (4 a b c d-7 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{9/2} d^{9/2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 349, 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 (4 a b c d-7 (a d+b 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^{9/2} d^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^3}{512 b^4 d^4}-\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^2}{768 b^4 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)}{192 b^4 d^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} \left (4 a b c d-7 (a d+b c)^2\right )}{96 b^3 d^2}-\frac {7 (a+b x)^{5/2} (c+d x)^{5/2} (a d+b c)}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{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)^{3/2} (c+d x)^{3/2} \, dx &=\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}+\frac {\int (a+b x)^{3/2} (c+d x)^{3/2} \left (-a c-\frac {7}{2} (b c+a d) x\right ) \, dx}{6 b d}\\ &=-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx}{24 b^2 d^2}\\ &=-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d) \left (4 a b c d-7 (b c+a d)^2\right )\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{64 b^3 d^2}\\ &=-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{384 b^4 d^2}\\ &=-\frac {(b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^4 d^3}-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}+\frac {\left ((b c-a d)^3 \left (4 a b c d-7 (b c+a d)^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^4 d^3}\\ &=\frac {(b c-a d)^3 \left (4 a b c d-7 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^4 d^4}-\frac {(b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^4 d^3}-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (4 a b c d-7 (b c+a d)^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^4 d^4}\\ &=\frac {(b c-a d)^3 \left (4 a b c d-7 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^4 d^4}-\frac {(b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^4 d^3}-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (4 a b c d-7 (b c+a 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^5 d^4}\\ &=\frac {(b c-a d)^3 \left (4 a b c d-7 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^4 d^4}-\frac {(b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^4 d^3}-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (4 a b c d-7 (b c+a 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^5 d^4}\\ &=\frac {(b c-a d)^3 \left (4 a b c d-7 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^4 d^4}-\frac {(b c-a d)^2 \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^4 d^3}-\frac {(b c-a d) \left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{192 b^4 d^2}-\frac {\left (4 a b c d-7 (b c+a d)^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {7 (b c+a d) (a+b x)^{5/2} (c+d x)^{5/2}}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d}-\frac {(b c-a d)^4 \left (4 a b c d-7 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{9/2} d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 306, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^5 d^5+5 a^4 b d^4 (47 c+14 d x)-2 a^3 b^2 d^3 \left (33 c^2+76 c d x+28 d^2 x^2\right )+6 a^2 b^3 d^2 \left (-11 c^3+6 c^2 d x+20 c d^2 x^2+8 d^3 x^3\right )+a b^4 d \left (235 c^4-152 c^3 d x+120 c^2 d^2 x^2+2336 c d^3 x^3+1664 d^4 x^4\right )+b^5 \left (-105 c^5+70 c^4 d x-56 c^3 d^2 x^2+48 c^2 d^3 x^3+1664 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^4 d^4}+\frac {(b c-a d)^4 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{512 b^{9/2} d^{9/2}} \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.
\(2(299)=598\).
time = 0.07, size = 1037, normalized size = 2.97
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (96 a^{2} b^{3} d^{5} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+96 b^{5} c^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-112 a^{3} b^{2} d^{5} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-112 b^{5} c^{3} d^{2} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+3328 a \,b^{4} d^{5} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+3328 b^{5} c \,d^{4} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-210 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} d^{5}-210 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{5}+140 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,d^{5} x +140 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{4} d x +470 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b c \,d^{4}-132 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{2} d^{3}-132 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{3} d^{2}+470 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{4} d -304 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c \,d^{4} x +72 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{2} d^{3} x -304 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{3} d^{2} x +105 \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}+105 \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}-270 \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}+135 \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}+60 \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}+135 \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}-270 \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 +4672 a \,b^{4} c \,d^{4} x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+240 a^{2} b^{3} c \,d^{4} x^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+240 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^{4} d^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \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 = 0.91, size = 890, normalized size = 2.55 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{6} c^{6} - 18 \, a b^{5} c^{5} d + 9 \, a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + 9 \, a^{4} b^{2} c^{2} d^{4} - 18 \, a^{5} b c d^{5} + 7 \, 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} - 105 \, b^{6} c^{5} d + 235 \, a b^{5} c^{4} d^{2} - 66 \, a^{2} b^{4} c^{3} d^{3} - 66 \, a^{3} b^{3} c^{2} d^{4} + 235 \, a^{4} b^{2} c d^{5} - 105 \, a^{5} b d^{6} + 1664 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (3 \, b^{6} c^{2} d^{4} + 146 \, a b^{5} c d^{5} + 3 \, a^{2} b^{4} d^{6}\right )} x^{3} - 8 \, {\left (7 \, b^{6} c^{3} d^{3} - 15 \, a b^{5} c^{2} d^{4} - 15 \, a^{2} b^{4} c d^{5} + 7 \, a^{3} b^{3} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{6} c^{4} d^{2} - 76 \, a b^{5} c^{3} d^{3} + 18 \, a^{2} b^{4} c^{2} d^{4} - 76 \, a^{3} b^{3} c d^{5} + 35 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{5} d^{5}}, -\frac {15 \, {\left (7 \, b^{6} c^{6} - 18 \, a b^{5} c^{5} d + 9 \, a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + 9 \, a^{4} b^{2} c^{2} d^{4} - 18 \, a^{5} b c d^{5} + 7 \, 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} - 105 \, b^{6} c^{5} d + 235 \, a b^{5} c^{4} d^{2} - 66 \, a^{2} b^{4} c^{3} d^{3} - 66 \, a^{3} b^{3} c^{2} d^{4} + 235 \, a^{4} b^{2} c d^{5} - 105 \, a^{5} b d^{6} + 1664 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (3 \, b^{6} c^{2} d^{4} + 146 \, a b^{5} c d^{5} + 3 \, a^{2} b^{4} d^{6}\right )} x^{3} - 8 \, {\left (7 \, b^{6} c^{3} d^{3} - 15 \, a b^{5} c^{2} d^{4} - 15 \, a^{2} b^{4} c d^{5} + 7 \, a^{3} b^{3} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{6} c^{4} d^{2} - 76 \, a b^{5} c^{3} d^{3} + 18 \, a^{2} b^{4} c^{2} d^{4} - 76 \, a^{3} b^{3} c d^{5} + 35 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{5} d^{5}}\right ] \end {gather*}
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 \begin {gather*} \int x^{2} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \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. 2032 vs.
\(2 (299) = 598\).
time = 1.42, size = 2032, normalized size = 5.82 \begin {gather*} \text {Too large to display} \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 )}^{3/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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