Optimal. Leaf size=258 \[ -\frac {3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{7/2}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]
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Rubi [A] time = 0.15, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \[ \frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac {3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{7/2}}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x (a+b x)^{3/2} (c+d x)^{3/2} \, dx &=\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {(b c+a d) \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx}{2 b d}\\ &=-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 \left (c^2-\frac {a^2 d^2}{b^2}\right )\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{16 d}\\ &=-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^2 (b c+a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{32 b^3 d}\\ &=-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}+\frac {\left (3 (b c-a d)^3 (b c+a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^3 d^2}\\ &=\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 (b c-a d)^4 (b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^3 d^3}\\ &=\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 (b c-a d)^4 (b c+a d)\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 )}{128 b^4 d^3}\\ &=\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {\left (3 (b c-a d)^4 (b c+a d)\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 )}{128 b^4 d^3}\\ &=\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^3}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d^2}-\frac {(b c-a d) (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3 d}-\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2 d}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d}-\frac {3 (b c-a d)^4 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 250, normalized size = 0.97 \[ \frac {b \sqrt {d} \sqrt {a+b x} (c+d x) \left (15 a^4 d^4-10 a^3 b d^3 (4 c+d x)+2 a^2 b^2 d^2 \left (9 c^2+13 c d x+4 d^2 x^2\right )+2 a b^3 d \left (-20 c^3+13 c^2 d x+136 c d^2 x^2+88 d^3 x^3\right )+b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )-15 (b c-a d)^{9/2} (a d+b c) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{640 b^4 d^{7/2} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 694, normalized size = 2.69 \[ \left [\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 40 \, a b^{4} c^{3} d^{2} + 18 \, a^{2} b^{3} c^{2} d^{3} - 40 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 176 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (b^{5} c^{2} d^{3} + 34 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{3} d^{2} - 13 \, a b^{4} c^{2} d^{3} - 13 \, a^{2} b^{3} c d^{4} + 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, b^{4} d^{4}}, \frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 40 \, a b^{4} c^{3} d^{2} + 18 \, a^{2} b^{3} c^{2} d^{3} - 40 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 176 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (b^{5} c^{2} d^{3} + 34 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{3} d^{2} - 13 \, a b^{4} c^{2} d^{3} - 13 \, a^{2} b^{3} c d^{4} + 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, b^{4} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.53, size = 1511, normalized size = 5.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 942, normalized size = 3.65 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-256 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{4} x^{4}+15 a^{5} 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 )-45 a^{4} b c \,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 )+30 a^{3} b^{2} c^{2} 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 )+30 a^{2} b^{3} c^{3} 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 )-45 a \,b^{4} c^{4} 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 )-352 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} d^{4} x^{3}+15 b^{5} c^{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 )-352 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c \,d^{3} x^{3}-16 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a^{2} b^{2} d^{4} x^{2}-544 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} c \,d^{3} x^{2}-16 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c^{2} d^{2} x^{2}+20 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,d^{4} x -52 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c \,d^{3} x -52 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{2} d^{2} x +20 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{3} d x -30 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4}+80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3}-36 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2}+80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d -30 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4}\right )}{1280 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{3} 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 \[ \int x\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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 \[ \int x \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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