Optimal. Leaf size=330 \[ -\frac {(d e-c f)^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{64 d^{7/2} f^{7/2}}+\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{32 d^3 f^2}+\frac {\sqrt {c+d x} \sqrt {e+f x} (d e-c f) \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{64 d^3 f^3}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f} \]
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Rubi [A] time = 0.30, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {951, 80, 50, 63, 217, 206} \begin {gather*} \frac {(c+d x)^{3/2} \sqrt {e+f x} \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{32 d^3 f^2}+\frac {\sqrt {c+d x} \sqrt {e+f x} (d e-c f) \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{64 d^3 f^3}-\frac {(d e-c f)^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{64 d^{7/2} f^{7/2}}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f} \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 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx &=\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}+\frac {\int \sqrt {c+d x} \sqrt {e+f x} \left (\frac {1}{2} \left (-5 c C d e-3 c^2 C f+8 A d^2 f\right )-\frac {1}{2} d (5 C d e+11 c C f-8 B d f) x\right ) \, dx}{4 d^2 f}\\ &=-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}+\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \int \sqrt {c+d x} \sqrt {e+f x} \, dx}{16 d^2 f^2}\\ &=\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{32 d^3 f^2}-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}+\frac {\left ((d e-c f) \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}} \, dx}{64 d^3 f^2}\\ &=\frac {(d e-c f) \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^3}+\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{32 d^3 f^2}-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\left ((d e-c f)^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{128 d^3 f^3}\\ &=\frac {(d e-c f) \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^3}+\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{32 d^3 f^2}-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\left ((d e-c f)^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{64 d^4 f^3}\\ &=\frac {(d e-c f) \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^3}+\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{32 d^3 f^2}-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {\left ((d e-c f)^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{64 d^4 f^3}\\ &=\frac {(d e-c f) \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^3}+\frac {\left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) (c+d x)^{3/2} \sqrt {e+f x}}{32 d^3 f^2}-\frac {(5 C d e+11 c C f-8 B d f) (c+d x)^{3/2} (e+f x)^{3/2}}{24 d^2 f^2}+\frac {C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}-\frac {(d e-c f)^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{7/2} f^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.72, size = 306, normalized size = 0.93 \begin {gather*} \frac {d \sqrt {f} \sqrt {c+d x} (e+f x) \left (8 d f \left (6 A d f (c f+d (e+2 f x))+B \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )\right )+C \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )\right )-3 (d e-c f)^{5/2} \sqrt {\frac {d (e+f x)}{d e-c f}} \sinh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right ) \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )}{192 d^4 f^{7/2} \sqrt {e+f x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.98, size = 643, normalized size = 1.95 \begin {gather*} \frac {(d e-c f)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right ) \left (-16 A d^2 f^2+8 B c d f^2+8 B d^2 e f-5 c^2 C f^2-6 c C d e f-5 C d^2 e^2\right )}{64 d^{7/2} f^{7/2}}+\frac {\sqrt {e+f x} (d e-c f)^2 \left (\frac {48 A d^5 f^2 (e+f x)^3}{(c+d x)^3}-\frac {48 A d^4 f^3 (e+f x)^2}{(c+d x)^2}-\frac {48 A d^3 f^4 (e+f x)}{c+d x}+48 A d^2 f^5-\frac {24 B d^5 e f (e+f x)^3}{(c+d x)^3}-\frac {24 B c d^4 f^2 (e+f x)^3}{(c+d x)^3}+\frac {88 B d^4 e f^2 (e+f x)^2}{(c+d x)^2}-\frac {40 B c d^3 f^3 (e+f x)^2}{(c+d x)^2}-\frac {40 B d^3 e f^3 (e+f x)}{c+d x}+\frac {88 B c d^2 f^4 (e+f x)}{c+d x}-24 B c d f^5-24 B d^2 e f^4+\frac {15 c^2 C d^3 f^2 (e+f x)^3}{(c+d x)^3}+\frac {73 c^2 C d^2 f^3 (e+f x)^2}{(c+d x)^2}-\frac {55 c^2 C d f^4 (e+f x)}{c+d x}+15 c^2 C f^5+\frac {15 C d^5 e^2 (e+f x)^3}{(c+d x)^3}-\frac {55 C d^4 e^2 f (e+f x)^2}{(c+d x)^2}+\frac {18 c C d^4 e f (e+f x)^3}{(c+d x)^3}+\frac {73 C d^3 e^2 f^2 (e+f x)}{c+d x}-\frac {66 c C d^3 e f^2 (e+f x)^2}{(c+d x)^2}-\frac {66 c C d^2 e f^3 (e+f x)}{c+d x}+18 c C d e f^4+15 C d^2 e^2 f^3\right )}{192 d^3 f^3 \sqrt {c+d x} \left (\frac {d (e+f x)}{c+d x}-f\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 840, normalized size = 2.55 \begin {gather*} \left [\frac {3 \, {\left (5 \, C d^{4} e^{4} - 4 \, {\left (C c d^{3} + 2 \, B d^{4}\right )} e^{3} f - 2 \, {\left (C c^{2} d^{2} - 4 \, B c d^{3} - 8 \, A d^{4}\right )} e^{2} f^{2} - 4 \, {\left (C c^{3} d - 2 \, B c^{2} d^{2} + 8 \, A c d^{3}\right )} e f^{3} + {\left (5 \, C c^{4} - 8 \, B c^{3} d + 16 \, A c^{2} d^{2}\right )} f^{4}\right )} \sqrt {d f} \log \left (8 \, d^{2} f^{2} x^{2} + d^{2} e^{2} + 6 \, c d e f + c^{2} f^{2} - 4 \, {\left (2 \, d f x + d e + c f\right )} \sqrt {d f} \sqrt {d x + c} \sqrt {f x + e} + 8 \, {\left (d^{2} e f + c d f^{2}\right )} x\right ) + 4 \, {\left (48 \, C d^{4} f^{4} x^{3} + 15 \, C d^{4} e^{3} f - {\left (7 \, C c d^{3} + 24 \, B d^{4}\right )} e^{2} f^{2} - {\left (7 \, C c^{2} d^{2} - 16 \, B c d^{3} - 48 \, A d^{4}\right )} e f^{3} + 3 \, {\left (5 \, C c^{3} d - 8 \, B c^{2} d^{2} + 16 \, A c d^{3}\right )} f^{4} + 8 \, {\left (C d^{4} e f^{3} + {\left (C c d^{3} + 8 \, B d^{4}\right )} f^{4}\right )} x^{2} - 2 \, {\left (5 \, C d^{4} e^{2} f^{2} - 2 \, {\left (C c d^{3} + 4 \, B d^{4}\right )} e f^{3} + {\left (5 \, C c^{2} d^{2} - 8 \, B c d^{3} - 48 \, A d^{4}\right )} f^{4}\right )} x\right )} \sqrt {d x + c} \sqrt {f x + e}}{768 \, d^{4} f^{4}}, \frac {3 \, {\left (5 \, C d^{4} e^{4} - 4 \, {\left (C c d^{3} + 2 \, B d^{4}\right )} e^{3} f - 2 \, {\left (C c^{2} d^{2} - 4 \, B c d^{3} - 8 \, A d^{4}\right )} e^{2} f^{2} - 4 \, {\left (C c^{3} d - 2 \, B c^{2} d^{2} + 8 \, A c d^{3}\right )} e f^{3} + {\left (5 \, C c^{4} - 8 \, B c^{3} d + 16 \, A c^{2} d^{2}\right )} f^{4}\right )} \sqrt {-d f} \arctan \left (\frac {{\left (2 \, d f x + d e + c f\right )} \sqrt {-d f} \sqrt {d x + c} \sqrt {f x + e}}{2 \, {\left (d^{2} f^{2} x^{2} + c d e f + {\left (d^{2} e f + c d f^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, C d^{4} f^{4} x^{3} + 15 \, C d^{4} e^{3} f - {\left (7 \, C c d^{3} + 24 \, B d^{4}\right )} e^{2} f^{2} - {\left (7 \, C c^{2} d^{2} - 16 \, B c d^{3} - 48 \, A d^{4}\right )} e f^{3} + 3 \, {\left (5 \, C c^{3} d - 8 \, B c^{2} d^{2} + 16 \, A c d^{3}\right )} f^{4} + 8 \, {\left (C d^{4} e f^{3} + {\left (C c d^{3} + 8 \, B d^{4}\right )} f^{4}\right )} x^{2} - 2 \, {\left (5 \, C d^{4} e^{2} f^{2} - 2 \, {\left (C c d^{3} + 4 \, B d^{4}\right )} e f^{3} + {\left (5 \, C c^{2} d^{2} - 8 \, B c d^{3} - 48 \, A d^{4}\right )} f^{4}\right )} x\right )} \sqrt {d x + c} \sqrt {f x + e}}{384 \, d^{4} f^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.33, size = 1103, normalized size = 3.34
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 = 1431, normalized size = 4.34
result too large to display
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 [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
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 \begin {gather*} \int \sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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