Optimal. Leaf size=369 \[ \frac {32 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^5 e^2 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^4 e^2}+\frac {12 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^3 e^2}+\frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{7 c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.55, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 5, 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)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{7 c^2 e^2 (2 c d-b e)}+\frac {12 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^3 e^2}+\frac {16 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^4 e^2}+\frac {32 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^5 e^2 \sqrt {d+e x}}+\frac {2 (d+e x)^{9/2} (-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.
[In]
[Out]
Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2} (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)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(7 c e f+9 c d g-8 b e g) \int \frac {(d+e x)^{7/2}}{\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)^{9/2}}{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 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac {(6 (7 c e f+9 c d g-8 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}{7 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {12 (7 c e f+9 c d g-8 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^3 e^2}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac {(24 (2 c d-b e) (7 c e f+9 c d g-8 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^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{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) (7 c e f+9 c d g-8 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^4 e^2}+\frac {12 (7 c e f+9 c d g-8 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^3 e^2}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac {\left (16 (2 c d-b e)^2 (7 c e f+9 c d g-8 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}{35 c^4 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e)^2 (7 c e f+9 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^5 e^2 \sqrt {d+e x}}+\frac {16 (2 c d-b e) (7 c e f+9 c d g-8 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^4 e^2}+\frac {12 (7 c e f+9 c d g-8 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^3 e^2}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 245, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-128 b^4 e^4 g+16 b^3 c e^3 (53 d g+7 e f-4 e g x)-8 b^2 c^2 e^2 \left (257 d^2 g+d e (77 f-45 g x)-e^2 x (7 f+2 g x)\right )-2 b c^3 e \left (-1075 d^3 g+d^2 e (334 g x-553 f)+d e^2 x (126 f+37 g x)+e^3 x^2 (7 f+4 g x)\right )+c^4 \left (-814 d^4 g+d^3 e (407 g x-637 f)+d^2 e^2 x (301 f+93 g x)+d e^3 x^2 (49 f+29 g x)+e^4 x^3 (7 f+5 g x)\right )\right )}{35 c^5 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 5.58, size = 418, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (-128 b^4 e^4 g-64 b^3 c e^3 g (d+e x)+912 b^3 c d e^3 g+112 b^3 c e^4 f-2400 b^2 c^2 d^2 e^2 g+56 b^2 c^2 e^3 f (d+e x)-672 b^2 c^2 d e^3 f+16 b^2 c^2 e^2 g (d+e x)^2+328 b^2 c^2 d e^2 g (d+e x)+2752 b c^3 d^3 e g+1344 b c^3 d^2 e^2 f-544 b c^3 d^2 e g (d+e x)-14 b c^3 e^2 f (d+e x)^2-224 b c^3 d e^2 f (d+e x)-8 b c^3 e g (d+e x)^3-50 b c^3 d e g (d+e x)^2-1152 c^4 d^4 g-896 c^4 d^3 e f+288 c^4 d^3 g (d+e x)+224 c^4 d^2 e f (d+e x)+36 c^4 d^2 g (d+e x)^2+7 c^4 e f (d+e x)^3+28 c^4 d e f (d+e x)^2+5 c^4 g (d+e x)^4+9 c^4 d g (d+e x)^3\right )}{35 c^5 e^2 \sqrt {d+e x} (b e+c (d+e x)-2 c d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 374, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (5 \, c^{4} e^{4} g x^{4} + {\left (7 \, c^{4} e^{4} f + {\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + {\left (7 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f + {\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 7 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f - 2 \, {\left (407 \, c^{4} d^{4} - 1075 \, b c^{3} d^{3} e + 1028 \, b^{2} c^{2} d^{2} e^{2} - 424 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g + {\left (7 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f + {\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, 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}}{35 \, {\left (c^{6} e^{4} x^{2} + b c^{5} e^{4} x - c^{6} d^{2} e^{2} + b c^{5} d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 367, normalized size = 0.99 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-5 g \,e^{4} x^{4} c^{4}+8 b \,c^{3} e^{4} g \,x^{3}-29 c^{4} d \,e^{3} g \,x^{3}-7 c^{4} e^{4} f \,x^{3}-16 b^{2} c^{2} e^{4} g \,x^{2}+74 b \,c^{3} d \,e^{3} g \,x^{2}+14 b \,c^{3} e^{4} f \,x^{2}-93 c^{4} d^{2} e^{2} g \,x^{2}-49 c^{4} d \,e^{3} f \,x^{2}+64 b^{3} c \,e^{4} g x -360 b^{2} c^{2} d \,e^{3} g x -56 b^{2} c^{2} e^{4} f x +668 b \,c^{3} d^{2} e^{2} g x +252 b \,c^{3} d \,e^{3} f x -407 c^{4} d^{3} e g x -301 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -848 b^{3} c d \,e^{3} g -112 b^{3} c \,e^{4} f +2056 b^{2} c^{2} d^{2} e^{2} g +616 b^{2} c^{2} d \,e^{3} f -2150 b \,c^{3} d^{3} e g -1106 b \,c^{3} d^{2} e^{2} f +814 c^{4} d^{4} g +637 f \,d^{3} c^{4} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{35 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} c^{5} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.74, size = 317, normalized size = 0.86 \begin {gather*} -\frac {2 \, {\left (c^{3} e^{3} x^{3} - 91 \, c^{3} d^{3} + 158 \, b c^{2} d^{2} e - 88 \, b^{2} c d e^{2} + 16 \, b^{3} e^{3} + {\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} + {\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{5 \, \sqrt {-c e x + c d - b e} c^{4} e} - \frac {2 \, {\left (5 \, c^{4} e^{4} x^{4} - 814 \, c^{4} d^{4} + 2150 \, b c^{3} d^{3} e - 2056 \, b^{2} c^{2} d^{2} e^{2} + 848 \, b^{3} c d e^{3} - 128 \, b^{4} e^{4} + {\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + {\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{35 \, \sqrt {-c e x + c d - b e} c^{5} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.09, size = 398, normalized size = 1.08 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,g\,x^4\,\sqrt {d+e\,x}}{7\,c^2}-\frac {\sqrt {d+e\,x}\,\left (256\,g\,b^4\,e^4-1696\,g\,b^3\,c\,d\,e^3-224\,f\,b^3\,c\,e^4+4112\,g\,b^2\,c^2\,d^2\,e^2+1232\,f\,b^2\,c^2\,d\,e^3-4300\,g\,b\,c^3\,d^3\,e-2212\,f\,b\,c^3\,d^2\,e^2+1628\,g\,c^4\,d^4+1274\,f\,c^4\,d^3\,e\right )}{35\,c^6\,e^4}+\frac {x^2\,\sqrt {d+e\,x}\,\left (32\,g\,b^2\,c^2\,e^4-148\,g\,b\,c^3\,d\,e^3-28\,f\,b\,c^3\,e^4+186\,g\,c^4\,d^2\,e^2+98\,f\,c^4\,d\,e^3\right )}{35\,c^6\,e^4}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (29\,c\,d\,g-8\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^3\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,g\,b^3\,c\,e^4+720\,g\,b^2\,c^2\,d\,e^3+112\,f\,b^2\,c^2\,e^4-1336\,g\,b\,c^3\,d^2\,e^2-504\,f\,b\,c^3\,d\,e^3+814\,g\,c^4\,d^3\,e+602\,f\,c^4\,d^2\,e^2\right )}{35\,c^6\,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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________