Optimal. Leaf size=268 \[ \frac {(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]
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Rubi [A] time = 0.16, antiderivative size = 268, 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} \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}+\frac {(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx &=\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(3 b c+7 a d) \int \sqrt {a+b x} (c+d x)^{5/2} \, dx}{10 b d}\\ &=-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {((b c-a d) (3 b c+7 a d)) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{16 b^2 d}\\ &=-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {\left ((b c-a d)^2 (3 b c+7 a d)\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{32 b^3 d}\\ &=-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {\left ((b c-a d)^3 (3 b c+7 a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^4 d}\\ &=-\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^4 (3 b c+7 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^4 d^2}\\ &=-\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^4 (3 b c+7 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^5 d^2}\\ &=-\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^4 (3 b c+7 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^5 d^2}\\ &=-\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {(b c-a d)^4 (3 b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.63, size = 291, normalized size = 1.09 \[ \frac {(a+b x)^{3/2} (c+d x)^{7/2} \left (3-\frac {(7 a d+3 b c) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 b^5 d^2 (a+b x)^2 (b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (15 a^2 d^2-10 a b d (5 c+2 d x)+b^2 \left (59 c^2+68 c d x+24 d^2 x^2\right )\right )+15 b^5 d (a+b x) (b c-a d)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}}-15 b^5 \sqrt {d} \sqrt {a+b x} (b c-a d)^5 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{128 b^9 d^2 (a+b x)^2 (c+d x)^4 \sqrt {b c-a d}}\right )}{15 b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 704, normalized size = 2.63 \[ \left [\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{5} d^{3}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{5} d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.83, size = 1522, normalized size = 5.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 942, normalized size = 3.51 \[ \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{4} x^{4}+105 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 )-375 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 )+450 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 )-150 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 )-75 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 )+96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} d^{4} x^{3}+45 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 )+2016 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c \,d^{3} x^{3}-112 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a^{2} b^{2} d^{4} x^{2}+352 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} c \,d^{3} x^{2}+1488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c^{2} d^{2} x^{2}+140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,d^{4} x -444 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c \,d^{3} x +436 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{2} d^{2} x +60 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{3} d x -210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4}+680 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3}-692 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2}+120 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d -90 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{2}} \]
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 \[ \int x\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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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