Optimal. Leaf size=574 \[ -\frac {\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (-b^3 e^3+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (-3 b^4 e^4-20 b^3 c d e^3+656 b^2 c^2 d^2 e^2-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (10 A c e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-3 b^5 e^5-20 b^4 c d e^4-240 b^3 c^2 d^2 e^3+1920 b^2 c^3 d^3 e^2-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac {d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 e^7}+\frac {\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.93, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {812, 814, 843, 620, 206, 724} \begin {gather*} -\frac {\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (48 b^2 c d e^2-b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (10 A c e \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac {d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 e^7}+\frac {\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 620
Rule 724
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac {\int \frac {(b (6 B d-5 A e)+(12 B c d-b B e-10 A c e) x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e^2}\\ &=-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\int \frac {\left (\frac {1}{2} b d \left (10 A c e (8 c d-7 b e)-2 B \left (48 c^2 d^2-46 b c d e+\frac {3 b^2 e^2}{2}\right )\right )+\frac {1}{2} \left (8 b c e (6 B d-5 A e) (2 c d-b e)-2 (12 B c d-b B e-10 A c e) \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}\right )\right ) x\right ) \sqrt {b x+c x^2}}{d+e x} \, dx}{16 c e^4}\\ &=-\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac {\int \frac {-\frac {1}{4} b d \left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )\right )-\frac {1}{4} \left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{64 c^2 e^6}\\ &=-\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\left (d^2 (c d-b e)^2 (B d (12 c d-7 b e)-5 A e (2 c d-b e))\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 e^7}+\frac {\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^2 e^7}\\ &=-\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac {\left (d^2 (c d-b e)^2 (B d (12 c d-7 b e)-5 A e (2 c d-b e))\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^7}+\frac {\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^2 e^7}\\ &=-\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{5/2} e^7}+\frac {d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 e^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.20, size = 618, normalized size = 1.08 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (\frac {e \sqrt {x} \left (10 A c e \left (15 b^3 e^3 (d+e x)+2 b^2 c e^2 \left (-360 d^2-205 d e x+59 e^2 x^2\right )+8 b c^2 e \left (210 d^3+110 d^2 e x-35 d e^2 x^2+17 e^3 x^3\right )-16 c^3 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )+B \left (-45 b^4 e^4 (d+e x)+30 b^3 c e^3 \left (-10 d^2-9 d e x+e^2 x^2\right )+8 b^2 c^2 e^2 \left (1230 d^3+695 d^2 e x-202 d e^2 x^2+93 e^3 x^3\right )+16 b c^3 e \left (-1320 d^4-690 d^3 e x+220 d^2 e^2 x^2-107 d e^3 x^3+63 e^4 x^4\right )+192 c^4 \left (60 d^5+30 d^4 e x-10 d^3 e^2 x^2+5 d^2 e^3 x^3-3 d e^4 x^4+2 e^5 x^5\right )\right )\right )}{d+e x}+\frac {1920 c^2 d^{3/2} (c d-b e)^{3/2} (5 A e (b e-2 c d)+B d (12 c d-7 b e)) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )+\frac {15 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (B \left (3 b^5 e^5+20 b^4 c d e^4+240 b^3 c^2 d^2 e^3-1920 b^2 c^3 d^3 e^2+3200 b c^4 d^4 e-1536 c^5 d^5\right )-10 A c e \left (b^4 e^4+16 b^3 c d e^3-144 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right )\right )}{\sqrt {b} \sqrt {\frac {c x}{b}+1}}\right )}{1920 c^{5/2} e^7 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 6.62, size = 852, normalized size = 1.48 \begin {gather*} \frac {\sqrt {c x^2+b x} \left (11520 B c^4 d^5-9600 A c^4 e d^4-21120 b B c^3 e d^4+5760 B c^4 e x d^4+16800 A b c^3 e^2 d^3+9840 b^2 B c^2 e^2 d^3-1920 B c^4 e^2 x^2 d^3-4800 A c^4 e^2 x d^3-11040 b B c^3 e^2 x d^3-7200 A b^2 c^2 e^3 d^2-300 b^3 B c e^3 d^2+960 B c^4 e^3 x^3 d^2+1600 A c^4 e^3 x^2 d^2+3520 b B c^3 e^3 x^2 d^2+8800 A b c^3 e^3 x d^2+5560 b^2 B c^2 e^3 x d^2-45 b^4 B e^4 d+150 A b^3 c e^4 d-576 B c^4 e^4 x^4 d-800 A c^4 e^4 x^3 d-1712 b B c^3 e^4 x^3 d-2800 A b c^3 e^4 x^2 d-1616 b^2 B c^2 e^4 x^2 d-4100 A b^2 c^2 e^4 x d-270 b^3 B c e^4 x d+384 B c^4 e^5 x^5+480 A c^4 e^5 x^4+1008 b B c^3 e^5 x^4+1360 A b c^3 e^5 x^3+744 b^2 B c^2 e^5 x^3+1180 A b^2 c^2 e^5 x^2+30 b^3 B c e^5 x^2-45 b^4 B e^5 x+150 A b^3 c e^5 x\right )}{1920 c^2 e^6 (d+e x)}+\frac {\left (12 B c^2 \sqrt {c d-b e} d^{9/2}-10 A c^2 e \sqrt {c d-b e} d^{7/2}-19 b B c e \sqrt {c d-b e} d^{7/2}+7 b^2 B e^2 \sqrt {c d-b e} d^{5/2}+15 A b c e^2 \sqrt {c d-b e} d^{5/2}-5 A b^2 e^3 \sqrt {c d-b e} d^{3/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} d+\sqrt {c} e x-e \sqrt {c x^2+b x}}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^7}+\frac {\left (1536 B d^5 c^5-1280 A d^4 e c^5+2560 A b d^3 e^2 c^4-3200 b B d^4 e c^4-1440 A b^2 d^2 e^3 c^3+1920 b^2 B d^3 e^2 c^3+160 A b^3 d e^4 c^2-240 b^3 B d^2 e^3 c^2+10 A b^4 e^5 c-20 b^4 B d e^4 c-3 b^5 B e^5\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x}\right )}{256 c^{5/2} e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 77.27, size = 3709, normalized size = 6.46
result too large to display
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 = 0.06, size = 7095, normalized size = 12.36 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \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 (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________