Optimal. Leaf size=176 \[ \frac {(b d-a e) \left (5 a^2 e^2-13 a b d e+11 b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x} \left (5 a^2 e^2-13 a b d e+11 b^2 d^2\right )}{b^3}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {2 \sqrt {a+b x} (d+e x)^{3/2} (4 b d-3 a e)}{b^2} \]
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Rubi [A] time = 0.17, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {951, 80, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (5 a^2 e^2-13 a b d e+11 b^2 d^2\right )}{b^3}+\frac {(b d-a e) \left (5 a^2 e^2-13 a b d e+11 b^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2} \sqrt {e}}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {2 \sqrt {a+b x} (d+e x)^{3/2} (4 b d-3 a e)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rule 951
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx &=\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {\int \frac {\sqrt {d+e x} \left (3 e (3 b d-2 a e) (5 b d+2 a e)+12 b e^2 (4 b d-3 a e) x\right )}{\sqrt {a+b x}} \, dx}{3 b^2 e}\\ &=\frac {2 (4 b d-3 a e) \sqrt {a+b x} (d+e x)^{3/2}}{b^2}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {\left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{b^2}\\ &=\frac {\left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{b^3}+\frac {2 (4 b d-3 a e) \sqrt {a+b x} (d+e x)^{3/2}}{b^2}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {\left ((b d-a e) \left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 b^3}\\ &=\frac {\left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{b^3}+\frac {2 (4 b d-3 a e) \sqrt {a+b x} (d+e x)^{3/2}}{b^2}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {\left ((b d-a e) \left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right )\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 )}{b^4}\\ &=\frac {\left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{b^3}+\frac {2 (4 b d-3 a e) \sqrt {a+b x} (d+e x)^{3/2}}{b^2}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {\left ((b d-a e) \left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right )\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 )}{b^4}\\ &=\frac {\left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{b^3}+\frac {2 (4 b d-3 a e) \sqrt {a+b x} (d+e x)^{3/2}}{b^2}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {(b d-a e) \left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 163, normalized size = 0.93 \begin {gather*} \frac {\sqrt {d+e x} \left (\sqrt {a+b x} \left (15 a^2 e^2-a b e (49 d+10 e x)+b^2 \left (57 d^2+32 d e x+8 e^2 x^2\right )\right )+\frac {3 \sqrt {b d-a e} \left (5 a^2 e^2-13 a b d e+11 b^2 d^2\right ) \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{\sqrt {e} \sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.67, size = 220, normalized size = 1.25 \begin {gather*} \frac {\sqrt {a+\frac {b (d+e x)}{e}-\frac {b d}{e}} \left (15 a^2 e^2 \sqrt {d+e x}-10 a b e (d+e x)^{3/2}-39 a b d e \sqrt {d+e x}+33 b^2 d^2 \sqrt {d+e x}+8 b^2 (d+e x)^{5/2}+16 b^2 d (d+e x)^{3/2}\right )}{3 b^3}-\frac {\sqrt {\frac {b}{e}} \left (-5 a^3 e^3+18 a^2 b d e^2-24 a b^2 d^2 e+11 b^3 d^3\right ) \log \left (\sqrt {a+\frac {b (d+e x)}{e}-\frac {b d}{e}}-\sqrt {\frac {b}{e}} \sqrt {d+e x}\right )}{b^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 414, normalized size = 2.35 \begin {gather*} \left [-\frac {3 \, {\left (11 \, b^{3} d^{3} - 24 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\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 (8 \, b^{3} e^{3} x^{2} + 57 \, b^{3} d^{2} e - 49 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{12 \, b^{4} e}, -\frac {3 \, {\left (11 \, b^{3} d^{3} - 24 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\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 (8 \, b^{3} e^{3} x^{2} + 57 \, b^{3} d^{2} e - 49 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{6 \, b^{4} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 441, normalized size = 2.51 \begin {gather*} -\frac {\frac {45 \, {\left (\frac {{\left (b^{2} d - a b e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} d^{2} {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} {\left | b \right |} e^{2}}{b^{2}} - \frac {15 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} d {\left | b \right |} e}{b^{3}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 392, normalized size = 2.23 \begin {gather*} -\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (15 a^{3} e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-54 a^{2} b d \,e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+72 a \,b^{2} d^{2} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-33 b^{3} d^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-16 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{2} e^{2} x^{2}+20 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b \,e^{2} x -64 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d e x -30 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} e^{2}+98 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b d e -114 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d^{2}\right )}{6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3}} \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 [B] time = 73.15, size = 1797, normalized size = 10.21
result too large to display
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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