Optimal. Leaf size=372 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt {d+e x}}-\frac {5 c \sqrt {2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.60, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {792, 662, 664, 660, 208} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt {d+e x}}-\frac {5 c \sqrt {2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 660
Rule 662
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {(3 c e f-11 c d g+4 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx}{4 e (2 c d-b e)}\\ &=\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c (3 c e f-11 c d g+4 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{8 e (2 c d-b e)}\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c (3 c e f-11 c d g+4 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{8 e}\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c (2 c d-b e) (3 c e f-11 c d g+4 b e g)) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {1}{4} (5 c (2 c d-b e) (3 c e f-11 c d g+4 b e g)) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {5 c \sqrt {2 c d-b e} (3 c e f-11 c d g+4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.25, size = 130, normalized size = 0.35 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{7/2} \left (\frac {c (d+e x)^2 (4 b e g-11 c d g+3 c e f) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac {7 d g}{e}-7 f\right )}{14 e (d+e x)^{11/2} (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 5.37, size = 360, normalized size = 0.97 \begin {gather*} \frac {\sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (-12 b^2 e^2 g (d+e x)+6 b^2 d e^2 g-6 b^2 e^3 f-24 b c d^2 e g-27 b c e^2 f (d+e x)+24 b c d e^2 f+75 b c d e g (d+e x)+56 b c e g (d+e x)^2+24 c^2 d^3 g-24 c^2 d^2 e f-102 c^2 d^2 g (d+e x)+54 c^2 d e f (d+e x)+24 c^2 e f (d+e x)^2-136 c^2 d g (d+e x)^2+8 c^2 g (d+e x)^3\right )}{12 e^2 (d+e x)^{5/2}}-\frac {5 \left (4 b^2 c e^2 g-19 b c^2 d e g+3 b c^2 e^2 f+22 c^3 d^2 g-6 c^3 d e f\right ) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{4 e^2 \sqrt {b e-2 c d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 959, normalized size = 2.58 \begin {gather*} \left [\frac {15 \, {\left (3 \, c^{2} d^{3} e f + {\left (3 \, c^{2} e^{4} f - {\left (11 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, c^{2} d e^{3} f - {\left (11 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (11 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (3 \, c^{2} d^{2} e^{2} f - {\left (11 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (8 \, c^{2} e^{3} g x^{3} + 8 \, {\left (3 \, c^{2} e^{3} f - 7 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (206 \, c^{2} d^{3} - 107 \, b c d^{2} e + 6 \, b^{2} d e^{2}\right )} g + {\left (3 \, {\left (34 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} f - {\left (350 \, c^{2} d^{2} e - 187 \, b c d e^{2} + 12 \, b^{2} 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}}{24 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}}, -\frac {15 \, {\left (3 \, c^{2} d^{3} e f + {\left (3 \, c^{2} e^{4} f - {\left (11 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, c^{2} d e^{3} f - {\left (11 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (11 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (3 \, c^{2} d^{2} e^{2} f - {\left (11 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - {\left (8 \, c^{2} e^{3} g x^{3} + 8 \, {\left (3 \, c^{2} e^{3} f - 7 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (206 \, c^{2} d^{3} - 107 \, b c d^{2} e + 6 \, b^{2} d e^{2}\right )} g + {\left (3 \, {\left (34 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} f - {\left (350 \, c^{2} d^{2} e - 187 \, b c d e^{2} + 12 \, b^{2} 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}}{12 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}}\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 [B] time = 0.08, size = 1189, normalized size = 3.20 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (60 b^{2} c \,e^{4} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-285 b \,c^{2} d \,e^{3} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+45 b \,c^{2} e^{4} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+330 c^{3} d^{2} e^{2} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-90 c^{3} d \,e^{3} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+120 b^{2} c d \,e^{3} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-570 b \,c^{2} d^{2} e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+90 b \,c^{2} d \,e^{3} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+660 c^{3} d^{3} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-180 c^{3} d^{2} e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+60 b^{2} c \,d^{2} e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-285 b \,c^{2} d^{3} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+45 b \,c^{2} d^{2} e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+330 c^{3} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-90 c^{3} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-8 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} g \,x^{3}-56 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+112 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-24 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+12 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} g x -187 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} g x +27 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} f x +350 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e g x -102 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} f -107 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,d^{2} e g +3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} f +206 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{3} g -54 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{12 \left (e x +d \right )^{\frac {5}{2}} \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \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]
________________________________________________________________________________________