Optimal. Leaf size=349 \[ -\frac {\left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{9/2} d^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^3}{512 b^4 d^4}-\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^2}{768 b^4 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)}{192 b^4 d^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} \left (4 a b c d-7 (a d+b c)^2\right )}{96 b^3 d^2}-\frac {7 (a+b x)^{5/2} (c+d x)^{5/2} (a d+b c)}{60 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d} \]
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Rubi [A] time = 0.31, 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 = {90, 80, 50, 63, 217, 206} \begin {gather*} \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 {7 (a+b x)^{5/2} (c+d x)^{5/2} (a d+b c)}{60 b^2 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 {\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 {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 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
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 ) \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^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 ) \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^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 = 1.86, size = 218, normalized size = 0.62 \begin {gather*} \frac {(a+b x)^{5/2} (c+d x)^{5/2} \left (\frac {25 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \left (\frac {3 (b c-a d)^{7/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{5/2} (a+b x)^{5/2} \sqrt {\frac {b (c+d x)}{b c-a d}}}+\frac {3 (a d-b c)^3}{d^2 (a+b x)^2}+\frac {2 (b c-a d)^2}{d (a+b x)}-8 a d+16 b (c+d x)+8 b c\right )}{128 b^2 (c+d x)^2}-35 (a d+b c)+50 b d x\right )}{300 b^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.75, size = 519, normalized size = 1.49 \begin {gather*} \frac {(b c-a d)^4 \left (7 a^2 d^2+10 a b c d+7 b^2 c^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}}-\frac {\sqrt {c+d x} (b c-a d)^4 \left (\frac {105 a^2 b^5 d^2 (c+d x)^5}{(a+b x)^5}-\frac {595 a^2 b^4 d^3 (c+d x)^4}{(a+b x)^4}-\frac {1686 a^2 b^3 d^4 (c+d x)^3}{(a+b x)^3}+\frac {1386 a^2 b^2 d^5 (c+d x)^2}{(a+b x)^2}-\frac {595 a^2 b d^6 (c+d x)}{a+b x}+105 a^2 d^7+\frac {105 b^7 c^2 (c+d x)^5}{(a+b x)^5}-\frac {595 b^6 c^2 d (c+d x)^4}{(a+b x)^4}+\frac {150 a b^6 c d (c+d x)^5}{(a+b x)^5}+\frac {1386 b^5 c^2 d^2 (c+d x)^3}{(a+b x)^3}-\frac {850 a b^5 c d^2 (c+d x)^4}{(a+b x)^4}-\frac {1686 b^4 c^2 d^3 (c+d x)^2}{(a+b x)^2}+\frac {1980 a b^4 c d^3 (c+d x)^3}{(a+b x)^3}-\frac {595 b^3 c^2 d^4 (c+d x)}{a+b x}+\frac {1980 a b^3 c d^4 (c+d x)^2}{(a+b x)^2}-\frac {850 a b^2 c d^5 (c+d x)}{a+b x}+150 a b c d^6+105 b^2 c^2 d^5\right )}{7680 b^4 d^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, 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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giac [B] time = 3.92, size = 2032, normalized size = 5.82
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 = 3.55 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (2560 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} d^{5} x^{5}+3328 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} d^{5} x^{4}+3328 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c \,d^{4} x^{4}+105 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 )-270 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 )+135 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 )+60 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 )+135 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 )+96 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} d^{5} x^{3}-270 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 )+4672 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c \,d^{4} x^{3}+105 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 )+96 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{2} d^{3} x^{3}-112 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} d^{5} x^{2}+240 \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}+240 \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}-112 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{3} d^{2} x^{2}+140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,d^{5} x -304 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c \,d^{4} x +72 \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 -304 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{3} d^{2} x +140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{4} d x -210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} d^{5}+470 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b c \,d^{4}-132 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{2} d^{3}-132 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{3} d^{2}+470 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{4} d -210 \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^{4} d^{4}} \end {gather*}
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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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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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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