Optimal. Leaf size=267 \[ -\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{315 c^4 e^2 (d+e x)^{3/2}}-\frac {8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{105 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{21 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 267, 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 {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{315 c^4 e^2 (d+e x)^{3/2}}-\frac {8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{105 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{21 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int (d+e x)^{3/2} (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)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}-\frac {\left (2 \left (\frac {3}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{9 c e^3}\\ &=-\frac {2 (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}+\frac {(4 (2 c d-b e) (3 c e f+c d g-2 b e g)) \int \sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{21 c^2 e}\\ &=-\frac {8 (2 c d-b e) (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 c^3 e^2 \sqrt {d+e x}}-\frac {2 (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}+\frac {\left (8 (2 c d-b e)^2 (3 c e f+c d g-2 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx}{105 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{315 c^4 e^2 (d+e x)^{3/2}}-\frac {8 (2 c d-b e) (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 c^3 e^2 \sqrt {d+e x}}-\frac {2 (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 179, normalized size = 0.67 \begin {gather*} \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (-16 b^3 e^3 g+24 b^2 c e^2 (4 d g+e (f+g x))-6 b c^2 e \left (31 d^2 g+d e (22 f+20 g x)+e^2 x (6 f+5 g x)\right )+c^3 \left (106 d^3 g+3 d^2 e (71 f+53 g x)+6 d e^2 x (27 f+20 g x)+5 e^3 x^2 (9 f+7 g x)\right )\right )}{315 c^4 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.45, size = 248, normalized size = 0.93 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2} \left (-16 b^3 e^3 g+24 b^2 c e^2 g (d+e x)+72 b^2 c d e^2 g+24 b^2 c e^3 f-96 b c^2 d^2 e g-36 b c^2 e^2 f (d+e x)-96 b c^2 d e^2 f-30 b c^2 e g (d+e x)^2-60 b c^2 d e g (d+e x)+32 c^3 d^3 g+96 c^3 d^2 e f+24 c^3 d^2 g (d+e x)+45 c^3 e f (d+e x)^2+72 c^3 d e f (d+e x)+35 c^3 g (d+e x)^3+15 c^3 d g (d+e x)^2\right )}{315 c^4 e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 352, normalized size = 1.32 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} e^{4} g x^{4} + 5 \, {\left (9 \, c^{4} e^{4} f + {\left (17 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, {\left (13 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} f + {\left (13 \, c^{4} d^{2} e^{2} + 10 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 3 \, {\left (71 \, c^{4} d^{3} e - 115 \, b c^{3} d^{2} e^{2} + 52 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} f - 2 \, {\left (53 \, c^{4} d^{4} - 146 \, b c^{3} d^{3} e + 141 \, b^{2} c^{2} d^{2} e^{2} - 56 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4}\right )} g + {\left (3 \, {\left (17 \, c^{4} d^{2} e^{2} + 22 \, b c^{3} d e^{3} - 4 \, b^{2} c^{2} e^{4}\right )} f - {\left (53 \, c^{4} d^{3} e - 93 \, b c^{3} d^{2} e^{2} + 48 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\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}}{315 \, {\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 235, normalized size = 0.88 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-35 g \,e^{3} x^{3} c^{3}+30 b \,c^{2} e^{3} g \,x^{2}-120 c^{3} d \,e^{2} g \,x^{2}-45 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +120 b \,c^{2} d \,e^{2} g x +36 b \,c^{2} e^{3} f x -159 c^{3} d^{2} e g x -162 c^{3} d \,e^{2} f x +16 b^{3} e^{3} g -96 b^{2} c d \,e^{2} g -24 b^{2} c \,e^{3} f +186 b \,c^{2} d^{2} e g +132 b \,c^{2} d \,e^{2} f -106 c^{3} d^{3} g -213 f \,d^{2} c^{3} e \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{315 \sqrt {e x +d}\, c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.82, size = 354, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (15 \, c^{3} e^{3} x^{3} - 71 \, c^{3} d^{3} + 115 \, b c^{2} d^{2} e - 52 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \, {\left (13 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} + {\left (17 \, c^{3} d^{2} e + 22 \, b c^{2} d e^{2} - 4 \, b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{105 \, {\left (c^{3} e^{2} x + c^{3} d e\right )}} + \frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} - 106 \, c^{4} d^{4} + 292 \, b c^{3} d^{3} e - 282 \, b^{2} c^{2} d^{2} e^{2} + 112 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \, {\left (17 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (13 \, c^{4} d^{2} e^{2} + 10 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} - {\left (53 \, c^{4} d^{3} e - 93 \, b c^{3} d^{2} e^{2} + 48 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{315 \, {\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]
________________________________________________________________________________________
mupad [B] time = 2.89, size = 337, normalized size = 1.26 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (b\,e\,g+17\,c\,d\,g+9\,c\,e\,f\right )}{63\,c}+\frac {2\,e\,g\,x^4\,\sqrt {d+e\,x}}{9}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (-2\,g\,b^2\,e^2+10\,g\,b\,c\,d\,e+3\,f\,b\,c\,e^2+13\,g\,c^2\,d^2+39\,f\,c^2\,d\,e\right )}{105\,c^2\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (-16\,g\,b^3\,e^3+96\,g\,b^2\,c\,d\,e^2+24\,f\,b^2\,c\,e^3-186\,g\,b\,c^2\,d^2\,e-132\,f\,b\,c^2\,d\,e^2+106\,g\,c^3\,d^3+213\,f\,c^3\,d^2\,e\right )}{315\,c^4\,e^3}+\frac {x\,\sqrt {d+e\,x}\,\left (16\,g\,b^3\,c\,e^4-96\,g\,b^2\,c^2\,d\,e^3-24\,f\,b^2\,c^2\,e^4+186\,g\,b\,c^3\,d^2\,e^2+132\,f\,b\,c^3\,d\,e^3-106\,g\,c^4\,d^3\,e+102\,f\,c^4\,d^2\,e^2\right )}{315\,c^4\,e^3}\right )}{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 \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________