Optimal. Leaf size=270 \[ -\frac {16 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d+e x}}-\frac {8 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 270, 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 = {794, 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+5 c d g+7 c e f)}{35 c^2 e^2}-\frac {8 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac {16 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d+e x}}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}-\frac {\left (2 \left (\frac {1}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{7 c e^3}\\ &=-\frac {2 (7 c e f+5 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}}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}+\frac {(4 (2 c d-b e) (7 c e f+5 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}{35 c^2 e}\\ &=-\frac {8 (2 c d-b e) (7 c e f+5 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}}{105 c^3 e^2}-\frac {2 (7 c e f+5 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}}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}+\frac {\left (8 (2 c d-b e)^2 (7 c e f+5 c d g-6 b e g)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (7 c e f+5 c d g-6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 c^4 e^2 \sqrt {d+e x}}-\frac {8 (2 c d-b e) (7 c e f+5 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}}{105 c^3 e^2}-\frac {2 (7 c e f+5 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}}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 181, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {d+e x} (b e-c d+c e x) \left (-48 b^3 e^3 g+8 b^2 c e^2 (32 d g+7 e f+3 e g x)-2 b c^2 e \left (219 d^2 g+2 d e (63 f+26 g x)+e^2 x (14 f+9 g x)\right )+c^3 \left (230 d^3 g+d^2 e (301 f+115 g x)+2 d e^2 x (49 f+30 g x)+3 e^3 x^2 (7 f+5 g x)\right )\right )}{105 c^4 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.31, size = 248, normalized size = 0.92 \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)+232 b^2 c d e^2 g+56 b^2 c e^3 f-352 b c^2 d^2 e g-28 b c^2 e^2 f (d+e x)-224 b c^2 d e^2 f-18 b c^2 e g (d+e x)^2-68 b c^2 d e g (d+e x)+160 c^3 d^3 g+224 c^3 d^2 e f+40 c^3 d^2 g (d+e x)+21 c^3 e f (d+e x)^2+56 c^3 d e f (d+e x)+15 c^3 g (d+e x)^3+15 c^3 d g (d+e x)^2\right )}{105 c^4 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 235, normalized size = 0.87 \begin {gather*} -\frac {2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f + 2 \, {\left (10 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \, {\left (115 \, c^{3} d^{3} - 219 \, b c^{2} d^{2} e + 128 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g + {\left (14 \, {\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (115 \, c^{3} d^{2} e - 104 \, 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}}{105 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 235, normalized size = 0.87 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-15 g \,e^{3} x^{3} c^{3}+18 b \,c^{2} e^{3} g \,x^{2}-60 c^{3} d \,e^{2} g \,x^{2}-21 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +104 b \,c^{2} d \,e^{2} g x +28 b \,c^{2} e^{3} f x -115 c^{3} d^{2} e g x -98 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -256 b^{2} c d \,e^{2} g -56 b^{2} c \,e^{3} f +438 b \,c^{2} d^{2} e g +252 b \,c^{2} d \,e^{2} f -230 c^{3} d^{3} g -301 f \,d^{2} c^{3} e \right ) \sqrt {e x +d}}{105 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.72, size = 319, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 43 \, c^{3} d^{3} + 79 \, b c^{2} d^{2} e - 44 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + {\left (11 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (29 \, c^{3} d^{2} e - 18 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} f}{15 \, \sqrt {-c e x + c d - b e} c^{3} e} + \frac {2 \, {\left (15 \, c^{4} e^{4} x^{4} - 230 \, c^{4} d^{4} + 668 \, b c^{3} d^{3} e - 694 \, b^{2} c^{2} d^{2} e^{2} + 304 \, b^{3} c d e^{3} - 48 \, b^{4} e^{4} + 3 \, {\left (15 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + {\left (55 \, c^{4} d^{2} e^{2} - 26 \, b c^{3} d e^{3} + 6 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (115 \, c^{4} d^{3} e - 219 \, b c^{3} d^{2} e^{2} + 128 \, b^{2} c^{2} d e^{3} - 24 \, b^{3} c e^{4}\right )} x\right )} g}{105 \, \sqrt {-c e x + c d - b e} c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.77, size = 246, normalized size = 0.91 \begin {gather*} -\frac {\left (\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{7\,c}+\frac {\sqrt {d+e\,x}\,\left (-96\,g\,b^3\,e^3+512\,g\,b^2\,c\,d\,e^2+112\,f\,b^2\,c\,e^3-876\,g\,b\,c^2\,d^2\,e-504\,f\,b\,c^2\,d\,e^2+460\,g\,c^3\,d^3+602\,f\,c^3\,d^2\,e\right )}{105\,c^4\,e^3}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (20\,c\,d\,g-6\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^2\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-208\,g\,b\,c^2\,d\,e^2-56\,f\,b\,c^2\,e^3+230\,g\,c^3\,d^2\,e+196\,f\,c^3\,d\,e^2\right )}{105\,c^4\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________