Optimal. Leaf size=292 \[ \frac {16 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d+e x}}+\frac {8 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.41, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac {8 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac {16 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d+e x}}+\frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(5 c e f+7 c d g-6 b e g) \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}-\frac {(4 (5 c e f+7 c d g-6 b e g)) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{5 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (5 c e f+7 c d g-6 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2}+\frac {2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}-\frac {(8 (2 c d-b e) (5 c e f+7 c d g-6 b e g)) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{15 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (2 c d-b e) (5 c e f+7 c d g-6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^4 e^2 \sqrt {d+e x}}+\frac {8 (5 c e f+7 c d g-6 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2}+\frac {2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 168, normalized size = 0.58 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (48 b^3 e^3 g-8 b^2 c e^2 (28 d g+5 e f-3 e g x)+2 b c^2 e \left (167 d^2 g+d e (70 f-44 g x)-e^2 x (10 f+3 g x)\right )+c^3 \left (-158 d^3 g+d^2 e (79 g x-115 f)+2 d e^2 x (25 f+8 g x)+e^3 x^2 (5 f+3 g x)\right )\right )}{15 c^4 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.15, size = 265, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (48 b^3 e^3 g+24 b^2 c e^2 g (d+e x)-248 b^2 c d e^2 g-40 b^2 c e^3 f+416 b c^2 d^2 e g-20 b c^2 e^2 f (d+e x)+160 b c^2 d e^2 f-6 b c^2 e g (d+e x)^2-76 b c^2 d e g (d+e x)-224 c^3 d^3 g-160 c^3 d^2 e f+56 c^3 d^2 g (d+e x)+5 c^3 e f (d+e x)^2+40 c^3 d e f (d+e x)+3 c^3 g (d+e x)^3+7 c^3 d g (d+e x)^2\right )}{15 c^4 e^2 \sqrt {d+e x} (b e+c (d+e x)-2 c d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 257, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (3 \, c^{3} e^{3} g x^{3} + {\left (5 \, c^{3} e^{3} f + 2 \, {\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} - 5 \, {\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f - 2 \, {\left (79 \, c^{3} d^{3} - 167 \, b c^{2} d^{2} e + 112 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g + {\left (10 \, {\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (c^{5} e^{4} x^{2} + b c^{4} e^{4} x - c^{5} d^{2} e^{2} + b c^{4} d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 235, normalized size = 0.80 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3 g \,e^{3} x^{3} c^{3}-6 b \,c^{2} e^{3} g \,x^{2}+16 c^{3} d \,e^{2} g \,x^{2}+5 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -88 b \,c^{2} d \,e^{2} g x -20 b \,c^{2} e^{3} f x +79 c^{3} d^{2} e g x +50 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -224 b^{2} c d \,e^{2} g -40 b^{2} c \,e^{3} f +334 b \,c^{2} d^{2} e g +140 b \,c^{2} d \,e^{2} f -158 c^{3} d^{3} g -115 f \,d^{2} c^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{15 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 203, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (c^{2} e^{2} x^{2} - 23 \, c^{2} d^{2} + 28 \, b c d e - 8 \, b^{2} e^{2} + 2 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, \sqrt {-c e x + c d - b e} c^{3} e} - \frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 158 \, c^{3} d^{3} + 334 \, b c^{2} d^{2} e - 224 \, b^{2} c d e^{2} + 48 \, b^{3} e^{3} + 2 \, {\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} x^{2} + {\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt {-c e x + c d - b e} c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 267, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (16\,c\,d\,g-6\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^3\,e^2}-\frac {\sqrt {d+e\,x}\,\left (-96\,g\,b^3\,e^3+448\,g\,b^2\,c\,d\,e^2+80\,f\,b^2\,c\,e^3-668\,g\,b\,c^2\,d^2\,e-280\,f\,b\,c^2\,d\,e^2+316\,g\,c^3\,d^3+230\,f\,c^3\,d^2\,e\right )}{15\,c^5\,e^4}+\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{5\,c^2\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-176\,g\,b\,c^2\,d\,e^2-40\,f\,b\,c^2\,e^3+158\,g\,c^3\,d^2\,e+100\,f\,c^3\,d\,e^2\right )}{15\,c^5\,e^4}\right )}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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