Optimal. Leaf size=176 \[ \frac {\left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{b^3}+\frac {2 (4 b d-3 a e) \sqrt {a+b x} (d+e x)^{3/2}}{b^2}+\frac {8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac {(b d-a e) \left (11 b^2 d^2-13 a b d e+5 a^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{7/2} \sqrt {e}} \]
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Rubi [A]
time = 0.11, 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 = {965, 81, 52, 65,
223, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rule 965
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 ) \text {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 ) \text {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 = 152, normalized size = 0.86 \begin {gather*} \frac {b \sqrt {a+b x} \sqrt {d+e 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 )-3 \sqrt {\frac {b}{e}} \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 ) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{e}} \sqrt {d+e x}\right )}{3 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs.
\(2(150)=300\).
time = 0.09, size = 392, normalized size = 2.23
method | result | size |
default | \(-\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (-16 b^{2} e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e b}+15 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e b}+a e +b d}{2 \sqrt {e b}}\right ) a^{3} e^{3}-54 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e b}+a e +b d}{2 \sqrt {e b}}\right ) a^{2} b d \,e^{2}+72 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e b}+a e +b d}{2 \sqrt {e b}}\right ) a \,b^{2} d^{2} e -33 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e b}+a e +b d}{2 \sqrt {e b}}\right ) b^{3} d^{3}+20 \sqrt {e b}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b \,e^{2} x -64 \sqrt {e b}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d e x -30 \sqrt {e b}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} e^{2}+98 \sqrt {e b}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b d e -114 \sqrt {e b}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d^{2}\right )}{6 b^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e b}}\) | \(392\) |
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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Fricas [A]
time = 2.48, size = 398, normalized size = 2.26 \begin {gather*} \left [-\frac {{\left (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^{\frac {1}{2}} \log \left (b^{2} d^{2} - 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, {\left (57 \, b^{3} d^{2} e + {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} e^{3} + {\left (32 \, b^{3} d x - 49 \, a b^{2} d\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{12 \, b^{4}}, -\frac {{\left (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 (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) - 2 \, {\left (57 \, b^{3} d^{2} e + {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} e^{3} + {\left (32 \, b^{3} d x - 49 \, a b^{2} d\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{6 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 441 vs.
\(2 (155) = 310\).
time = 4.35, 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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Mupad [B]
time = 73.15, size = 1797, normalized size = 10.21 \begin {gather*} \frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (110\,a^2\,d\,e^2+460\,a\,b\,d^2\,e+70\,b^2\,d^3\right )}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (-30\,a^2\,b\,d\,e^2+20\,a\,b^2\,d^2\,e+10\,b^3\,d^3\right )}{e^4\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {160\,\sqrt {a}\,d^{5/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (-30\,a^2\,d\,e^2+20\,a\,b\,d^2\,e+10\,b^2\,d^3\right )}{b^2\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (110\,a^2\,d\,e^2+460\,a\,b\,d^2\,e+70\,b^2\,d^3\right )}{b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}-\frac {\sqrt {a}\,\sqrt {d}\,\left (320\,b\,d^2+640\,a\,e\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {160\,\sqrt {a}\,b^2\,d^{5/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {b^4}{e^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (-10\,a^3\,b^2\,e^3+6\,a^2\,b^3\,d\,e^2+2\,a\,b^4\,d^2\,e+2\,b^5\,d^3\right )}{e^6\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (132\,a^3\,e^3+1252\,a^2\,b\,d\,e^2+1100\,a\,b^2\,d^2\,e+76\,b^3\,d^3\right )}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (-\frac {170\,a^3\,b\,e^3}{3}+34\,a^2\,b^2\,d\,e^2+182\,a\,b^3\,d^2\,e+\frac {34\,b^4\,d^3}{3}\right )}{e^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (-10\,a^3\,e^3+6\,a^2\,b\,d\,e^2+2\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{b^3\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (-\frac {170\,a^3\,e^3}{3}+34\,a^2\,b\,d\,e^2+182\,a\,b^2\,d^2\,e+\frac {34\,b^3\,d^3}{3}\right )}{b^2\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^9}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (132\,a^3\,e^3+1252\,a^2\,b\,d\,e^2+1100\,a\,b^2\,d^2\,e+76\,b^3\,d^3\right )}{b\,e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}+\frac {\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (1024\,a^2\,e^2+\frac {5632\,a\,b\,d\,e}{3}+512\,b^2\,d^2\right )}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}+\frac {\sqrt {a}\,\sqrt {d}\,\left (256\,b\,d^2+768\,a\,e\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {\sqrt {a}\,\sqrt {d}\,\left (256\,b^3\,d^2+768\,a\,e\,b^2\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{12}}+\frac {b^6}{e^6}-\frac {6\,b^5\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}-\frac {6\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}}+\frac {\frac {\left (30\,b\,d^3+30\,a\,e\,d^2\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{e^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {120\,\sqrt {a}\,d^{5/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {\left (30\,b\,d^3+30\,a\,e\,d^2\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}+\frac {b^2}{e^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )\,\left (5\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2\right )}{b^{7/2}\,\sqrt {e}}-\frac {30\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )}{b^{3/2}\,\sqrt {e}}+\frac {10\,d\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )\,\left (3\,a\,e+b\,d\right )}{b^{5/2}\,\sqrt {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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