Optimal. Leaf size=200 \[ -\frac {8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac {8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt {d+e x}}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {870, 794, 648} \begin {gather*} \frac {8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt {d+e x}}-\frac {8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx &=\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}+\frac {\left (4 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{7 c d e^2}\\ &=\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 c^2 d^2 e \sqrt {d+e x}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}-\frac {\left (4 (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{35 c^2 d^2 e}\\ &=-\frac {8 (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 c^2 d^2 e \sqrt {d+e x}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 90, normalized size = 0.45 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (8 a^2 e^2 g^2-4 a c d e g (7 f+3 g x)+c^2 d^2 \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )}{105 c^3 d^3 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 365, normalized size = 1.82 \begin {gather*} \frac {2 \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}} \left (8 a^3 e^6 g^2+4 a^2 c d^2 e^4 g^2-28 a^2 c d e^5 f g-4 a^2 c d e^4 g^2 (d+e x)+3 a c^2 d^4 e^2 g^2-14 a c^2 d^3 e^3 f g-6 a c^2 d^3 e^2 g^2 (d+e x)+35 a c^2 d^2 e^4 f^2+14 a c^2 d^2 e^3 f g (d+e x)+3 a c^2 d^2 e^2 g^2 (d+e x)^2-15 c^3 d^6 g^2+42 c^3 d^5 e f g+45 c^3 d^5 g^2 (d+e x)-35 c^3 d^4 e^2 f^2-84 c^3 d^4 e f g (d+e x)-45 c^3 d^4 g^2 (d+e x)^2+35 c^3 d^3 e^2 f^2 (d+e x)+42 c^3 d^3 e f g (d+e x)^2+15 c^3 d^3 g^2 (d+e x)^3\right )}{105 c^3 d^3 e^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 173, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (15 \, c^{3} d^{3} g^{2} x^{3} + 35 \, a c^{2} d^{2} e f^{2} - 28 \, a^{2} c d e^{2} f g + 8 \, a^{3} e^{3} g^{2} + 3 \, {\left (14 \, c^{3} d^{3} f g + a c^{2} d^{2} e g^{2}\right )} x^{2} + {\left (35 \, c^{3} d^{3} f^{2} + 14 \, a c^{2} d^{2} e f g - 4 \, a^{2} c d e^{2} g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{105 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{2}}{\sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 0.58 \begin {gather*} \frac {2 \left (c d x +a e \right ) \left (15 g^{2} x^{2} c^{2} d^{2}-12 a c d e \,g^{2} x +42 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-28 a c d e f g +35 f^{2} c^{2} d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \sqrt {e x +d}\, c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 133, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f^{2}}{3 \, c d} + \frac {4 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} f g}{15 \, c^{2} d^{2}} + \frac {2 \, {\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} g^{2}}{105 \, c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.25, size = 157, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^2\,x^3}{7}+\frac {16\,a^3\,e^3\,g^2-56\,a^2\,c\,d\,e^2\,f\,g+70\,a\,c^2\,d^2\,e\,f^2}{105\,c^3\,d^3}+\frac {x\,\left (-8\,a^2\,c\,d\,e^2\,g^2+28\,a\,c^2\,d^2\,e\,f\,g+70\,c^3\,d^3\,f^2\right )}{105\,c^3\,d^3}+\frac {2\,g\,x^2\,\left (a\,e\,g+14\,c\,d\,f\right )}{35\,c\,d}\right )}{\sqrt {d+e\,x}} \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 \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{2}}{\sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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