Optimal. Leaf size=250 \[ \frac {(b d-a e)^3 (5 a B e-8 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (5 a B e-8 A b e+3 b B d)}{64 b^3 e^2}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (5 a B e-8 A b e+3 b B d)}{32 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (5 a B e-8 A b e+3 b B d)}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e} \]
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Rubi [A] time = 0.20, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ -\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (5 a B e-8 A b e+3 b B d)}{64 b^3 e^2}+\frac {(b d-a e)^3 (5 a B e-8 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (5 a B e-8 A b e+3 b B d)}{32 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (5 a B e-8 A b e+3 b B d)}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx &=\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {3 b d}{2}+\frac {5 a e}{2}\right )\right ) \int \sqrt {a+b x} (d+e x)^{3/2} \, dx}{4 b e}\\ &=-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}-\frac {((b d-a e) (3 b B d-8 A b e+5 a B e)) \int \sqrt {a+b x} \sqrt {d+e x} \, dx}{16 b^2 e}\\ &=-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}-\frac {\left ((b d-a e)^2 (3 b B d-8 A b e+5 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{64 b^3 e}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b^3 e^2}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^4 e^2}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{64 b^4 e^2}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 2.22, size = 266, normalized size = 1.06 \[ \frac {\sqrt {b d-a e} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} (-5 a B e+8 A b e-3 b B d) \left (2 b^4 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} (-3 a e+7 b d+4 b e x)+3 b^4 e (a+b x) (b d-a e)^{7/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-3 b^4 \sqrt {e} \sqrt {a+b x} (b d-a e)^4 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )+48 b^8 B e^2 (a+b x)^2 (d+e x)^4}{192 b^9 e^3 \sqrt {a+b x} (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 766, normalized size = 3.06 \[ \left [\frac {3 \, {\left (3 \, B b^{4} d^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (48 \, B b^{4} e^{4} x^{3} - 9 \, B b^{4} d^{3} e + 3 \, {\left (3 \, B a b^{3} + 8 \, A b^{4}\right )} d^{2} e^{2} - {\left (31 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} + 3 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (9 \, B b^{4} d e^{3} + {\left (B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (3 \, B b^{4} d^{2} e^{2} + 2 \, {\left (5 \, B a b^{3} + 28 \, A b^{4}\right )} d e^{3} - {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{4} e^{3}}, -\frac {3 \, {\left (3 \, B b^{4} d^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, B b^{4} e^{4} x^{3} - 9 \, B b^{4} d^{3} e + 3 \, {\left (3 \, B a b^{3} + 8 \, A b^{4}\right )} d^{2} e^{2} - {\left (31 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} + 3 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (9 \, B b^{4} d e^{3} + {\left (B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (3 \, B b^{4} d^{2} e^{2} + 2 \, {\left (5 \, B a b^{3} + 28 \, A b^{4}\right )} d e^{3} - {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{4} e^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.71, size = 1386, normalized size = 5.54 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1150, normalized size = 4.60 \[ \frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (24 A \,a^{3} b \,e^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-72 A \,a^{2} b^{2} d \,e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+72 A a \,b^{3} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-24 A \,b^{4} d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-15 B \,a^{4} e^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+36 B \,a^{3} b d \,e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-18 B \,a^{2} b^{2} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-12 B a \,b^{3} d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+9 B \,b^{4} d^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+96 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}\, B \,b^{3} e^{3} x^{3}+128 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}\, A \,b^{3} e^{3} x^{2}+16 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}\, B a \,b^{2} e^{3} x^{2}+144 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}\, B \,b^{3} d \,e^{2} x^{2}+32 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, A a \,b^{2} e^{3} x +224 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, A \,b^{3} d \,e^{2} x -20 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B \,a^{2} b \,e^{3} x +40 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B a \,b^{2} d \,e^{2} x +12 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B \,b^{3} d^{2} e x -48 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, A \,a^{2} b \,e^{3}+128 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}\, A a \,b^{2} d \,e^{2}+48 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, A \,b^{3} d^{2} e +30 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B \,a^{3} e^{3}-62 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B \,a^{2} b d \,e^{2}+18 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B a \,b^{2} d^{2} e -18 \sqrt {b e}\, \sqrt {b e \,x^{2}+a e x +b d x +a d}\, B \,b^{3} d^{3}\right )}{384 \sqrt {b e \,x^{2}+a e x +b d x +a d}\, \sqrt {b e}\, b^{3} e^{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 \[ \int \left (A+B\,x\right )\,\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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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