Optimal. Leaf size=540 \[ -\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f} \]
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Rubi [A] time = 0.71, antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1615, 147, 50, 63, 217, 206} \[ -\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+5 c^3 f^3+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+5 c^3 f^3+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 147
Rule 206
Rule 217
Rule 1615
Rubi steps
\begin {align*} \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx &=\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}+\frac {\int \frac {(a+b x) \sqrt {c+d x} \left (-\frac {1}{2} b (4 b c C e+3 a C d e+a c C f-8 A b d f)-\frac {1}{2} b (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{\sqrt {e+f x}} \, dx}{4 b^2 d f}\\ &=\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}} \, dx}{64 d^3 f^3}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {\left ((d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{128 d^3 f^4}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {\left ((d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\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^4}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {\left ((d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\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^4}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {(d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{7/2} f^{9/2}}\\ \end {align*}
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Mathematica [A] time = 3.54, size = 478, normalized size = 0.89 \[ \frac {d \sqrt {f} \sqrt {c+d x} (e+f x) \left (8 a d f \left (6 d f (4 A d f+B (c f-3 d e+2 d f x))+C \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+b \left (8 d f \left (6 A d f (c f-3 d e+2 d f x)+B \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+C \left (15 c^3 f^3+c^2 d f^2 (17 e-10 f x)+c d^2 f \left (25 e^2-12 e f x+8 f^2 x^2\right )+d^3 \left (-105 e^3+70 e^2 f x-56 e f^2 x^2+48 f^3 x^3\right )\right )\right )\right )+3 (d e-c f)^{3/2} \sqrt {\frac {d (e+f x)}{d e-c f}} \sinh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right ) \left (b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-8 a d f \left (2 d f (4 A d f-B c f-3 B d e)+C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )}{192 d^4 f^{9/2} \sqrt {e+f x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.38, size = 1114, normalized size = 2.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.82, size = 736, normalized size = 1.36 \[ \frac {{\left (\sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )} C b}{d^{4} f} - \frac {17 \, C b c d^{12} f^{6} - 8 \, C a d^{13} f^{6} - 8 \, B b d^{13} f^{6} + 7 \, C b d^{13} f^{5} e}{d^{16} f^{7}}\right )} + \frac {59 \, C b c^{2} d^{12} f^{6} - 56 \, C a c d^{13} f^{6} - 56 \, B b c d^{13} f^{6} + 48 \, B a d^{14} f^{6} + 48 \, A b d^{14} f^{6} + 50 \, C b c d^{13} f^{5} e - 40 \, C a d^{14} f^{5} e - 40 \, B b d^{14} f^{5} e + 35 \, C b d^{14} f^{4} e^{2}}{d^{16} f^{7}}\right )} - \frac {3 \, {\left (5 \, C b c^{3} d^{12} f^{6} - 8 \, C a c^{2} d^{13} f^{6} - 8 \, B b c^{2} d^{13} f^{6} + 16 \, B a c d^{14} f^{6} + 16 \, A b c d^{14} f^{6} - 64 \, A a d^{15} f^{6} + 9 \, C b c^{2} d^{13} f^{5} e - 16 \, C a c d^{14} f^{5} e - 16 \, B b c d^{14} f^{5} e + 48 \, B a d^{15} f^{5} e + 48 \, A b d^{15} f^{5} e + 15 \, C b c d^{14} f^{4} e^{2} - 40 \, C a d^{15} f^{4} e^{2} - 40 \, B b d^{15} f^{4} e^{2} + 35 \, C b d^{15} f^{3} e^{3}\right )}}{d^{16} f^{7}}\right )} \sqrt {d x + c} + \frac {3 \, {\left (5 \, C b c^{4} f^{4} - 8 \, C a c^{3} d f^{4} - 8 \, B b c^{3} d f^{4} + 16 \, B a c^{2} d^{2} f^{4} + 16 \, A b c^{2} d^{2} f^{4} - 64 \, A a c d^{3} f^{4} + 4 \, C b c^{3} d f^{3} e - 8 \, C a c^{2} d^{2} f^{3} e - 8 \, B b c^{2} d^{2} f^{3} e + 32 \, B a c d^{3} f^{3} e + 32 \, A b c d^{3} f^{3} e + 64 \, A a d^{4} f^{3} e + 6 \, C b c^{2} d^{2} f^{2} e^{2} - 24 \, C a c d^{3} f^{2} e^{2} - 24 \, B b c d^{3} f^{2} e^{2} - 48 \, B a d^{4} f^{2} e^{2} - 48 \, A b d^{4} f^{2} e^{2} + 20 \, C b c d^{3} f e^{3} + 40 \, C a d^{4} f e^{3} + 40 \, B b d^{4} f e^{3} - 35 \, C b d^{4} e^{4}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt {d f} d^{3} f^{4}}\right )} d}{192 \, {\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 2002, normalized size = 3.71 \[ \text {result too large to display} \]
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(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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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