Optimal. Leaf size=333 \[ -\frac {\sqrt {d+e x} \sqrt {f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac {(d+e x)^{3/2} \sqrt {f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac {(d+e x)^{5/2} \sqrt {f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g} \]
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Rubi [A] time = 0.35, antiderivative size = 333, 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} \[ \frac {(d+e x)^{3/2} \sqrt {f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}-\frac {\sqrt {d+e x} \sqrt {f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac {(d+e x)^{5/2} \sqrt {f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g} \]
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 {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx &=\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (8 a e^2 g-c d (7 e f+d g)\right )-\frac {1}{2} e (7 c e f+9 c d g-8 b e g) x\right )}{\sqrt {f+g x}} \, dx}{4 e^2 g}\\ &=-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x}} \, dx}{48 e^2 g^2}\\ &=\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}-\frac {\left ((e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x}} \, dx}{64 e^2 g^3}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{128 e^2 g^4}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{64 e^3 g^4}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{64 e^3 g^4}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {(e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{64 e^{5/2} g^{9/2}}\\ \end {align*}
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Mathematica [A] time = 1.54, size = 313, normalized size = 0.94 \[ \frac {3 (e f-d g)^{5/2} \sqrt {\frac {e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )-e \sqrt {g} \sqrt {d+e x} (f+g x) \left (c \left (9 d^3 g^3+3 d^2 e g^2 (5 f-2 g x)+d e^2 g \left (-145 f^2+92 f g x-72 g^2 x^2\right )+e^3 \left (105 f^3-70 f^2 g x+56 f g^2 x^2-48 g^3 x^3\right )\right )-8 e g \left (6 a e g (5 d g-3 e f+2 e g x)+b \left (3 d^2 g^2+2 d e g (7 g x-11 f)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )\right )}{192 e^3 g^{9/2} \sqrt {f+g x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 852, normalized size = 2.56 \[ \left [\frac {3 \, {\left (35 \, c e^{4} f^{4} - 20 \, {\left (3 \, c d e^{3} + 2 \, b e^{4}\right )} f^{3} g + 6 \, {\left (3 \, c d^{2} e^{2} + 12 \, b d e^{3} + 8 \, a e^{4}\right )} f^{2} g^{2} + 4 \, {\left (c d^{3} e - 6 \, b d^{2} e^{2} - 24 \, a d e^{3}\right )} f g^{3} + {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} g^{4}\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) + 4 \, {\left (48 \, c e^{4} g^{4} x^{3} - 105 \, c e^{4} f^{3} g + 5 \, {\left (29 \, c d e^{3} + 24 \, b e^{4}\right )} f^{2} g^{2} - {\left (15 \, c d^{2} e^{2} + 176 \, b d e^{3} + 144 \, a e^{4}\right )} f g^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} g^{4} - 8 \, {\left (7 \, c e^{4} f g^{3} - {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} g^{4}\right )} x^{2} + 2 \, {\left (35 \, c e^{4} f^{2} g^{2} - 2 \, {\left (23 \, c d e^{3} + 20 \, b e^{4}\right )} f g^{3} + {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} g^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{768 \, e^{3} g^{5}}, -\frac {3 \, {\left (35 \, c e^{4} f^{4} - 20 \, {\left (3 \, c d e^{3} + 2 \, b e^{4}\right )} f^{3} g + 6 \, {\left (3 \, c d^{2} e^{2} + 12 \, b d e^{3} + 8 \, a e^{4}\right )} f^{2} g^{2} + 4 \, {\left (c d^{3} e - 6 \, b d^{2} e^{2} - 24 \, a d e^{3}\right )} f g^{3} + {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} g^{4}\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c e^{4} g^{4} x^{3} - 105 \, c e^{4} f^{3} g + 5 \, {\left (29 \, c d e^{3} + 24 \, b e^{4}\right )} f^{2} g^{2} - {\left (15 \, c d^{2} e^{2} + 176 \, b d e^{3} + 144 \, a e^{4}\right )} f g^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} g^{4} - 8 \, {\left (7 \, c e^{4} f g^{3} - {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} g^{4}\right )} x^{2} + 2 \, {\left (35 \, c e^{4} f^{2} g^{2} - 2 \, {\left (23 \, c d e^{3} + 20 \, b e^{4}\right )} f g^{3} + {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} g^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{384 \, e^{3} g^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 448, normalized size = 1.35 \[ \frac {1}{192} \, \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}} {\left (2 \, {\left (4 \, {\left (x e + d\right )} {\left (\frac {6 \, {\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac {{\left (9 \, c d g^{6} e^{6} + 7 \, c f g^{5} e^{7} - 8 \, b g^{6} e^{7}\right )} e^{\left (-9\right )}}{g^{7}}\right )} + \frac {{\left (3 \, c d^{2} g^{6} e^{6} + 10 \, c d f g^{5} e^{7} - 8 \, b d g^{6} e^{7} + 35 \, c f^{2} g^{4} e^{8} - 40 \, b f g^{5} e^{8} + 48 \, a g^{6} e^{8}\right )} e^{\left (-9\right )}}{g^{7}}\right )} {\left (x e + d\right )} + \frac {3 \, {\left (3 \, c d^{3} g^{6} e^{6} + 7 \, c d^{2} f g^{5} e^{7} - 8 \, b d^{2} g^{6} e^{7} + 25 \, c d f^{2} g^{4} e^{8} - 32 \, b d f g^{5} e^{8} + 48 \, a d g^{6} e^{8} - 35 \, c f^{3} g^{3} e^{9} + 40 \, b f^{2} g^{4} e^{9} - 48 \, a f g^{5} e^{9}\right )} e^{\left (-9\right )}}{g^{7}}\right )} \sqrt {x e + d} - \frac {{\left (3 \, c d^{4} g^{4} + 4 \, c d^{3} f g^{3} e - 8 \, b d^{3} g^{4} e + 18 \, c d^{2} f^{2} g^{2} e^{2} - 24 \, b d^{2} f g^{3} e^{2} + 48 \, a d^{2} g^{4} e^{2} - 60 \, c d f^{3} g e^{3} + 72 \, b d f^{2} g^{2} e^{3} - 96 \, a d f g^{3} e^{3} + 35 \, c f^{4} e^{4} - 40 \, b f^{3} g e^{4} + 48 \, a f^{2} g^{2} e^{4}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} + \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{64 \, g^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1207, normalized size = 3.62 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )}{\sqrt {f+g\,x}} \,d x \]
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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