Optimal. Leaf size=379 \[ \frac {3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}-\frac {3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \]
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Rubi [A] time = 0.51, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {832, 779, 612, 621, 206} \begin {gather*} -\frac {3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac {3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\int (d+e x)^2 (3 c (b d-2 a e)+3 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\int (d+e x) \left (\frac {3}{2} c \left (5 b^2 d e-36 a c d e+4 b \left (c d^2+a e^2\right )\right )+\frac {3}{2} c \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{56 c^2}\\ &=\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {\left (\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{128 c^3}\\ &=\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}-\frac {\left (3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{2048 c^4}\\ &=-\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16384 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8192 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}+\frac {\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac {3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac {1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac {\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac {3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 297, normalized size = 0.78 \begin {gather*} \frac {1}{8} \left (\frac {(a+x (b+c x))^{5/2} \left (-8 c^2 e (a e (96 d+35 e x)+b d (13 d+10 e x))+2 b c e^2 (122 a e+112 b d+45 b e x)-63 b^3 e^3+16 c^3 d^2 (6 d+5 e x)\right )}{280 c^3}+\frac {e \left (b^2-4 a c\right ) \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2048 c^{11/2}}+2 (d+e x)^3 (a+x (b+c x))^{5/2}+\frac {3 (d+e x)^2 (a+x (b+c x))^{5/2} (2 c d-b e)}{7 c}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 4.92, size = 928, normalized size = 2.45 \begin {gather*} \frac {\sqrt {c x^2+b x+a} \left (-945 e^3 b^7+3360 c d e^2 b^6+630 c e^3 x b^6+10500 a c e^3 b^5-504 c^2 e^3 x^2 b^5-3360 c^2 d^2 e b^5-2240 c^2 d e^2 x b^5+432 c^3 e^3 x^3 b^4-35840 a c^2 d e^2 b^4+1792 c^3 d e^2 x^2 b^4-6328 a c^2 e^3 x b^4+2240 c^3 d^2 e x b^4-384 c^4 e^3 x^4 b^3-37744 a^2 c^2 e^3 b^3-1536 c^4 d e^2 x^3 b^3+4544 a c^3 e^3 x^2 b^3-1792 c^4 d^2 e x^2 b^3+35840 a c^3 d^2 e b^3+21504 a c^3 d e^2 x b^3+52480 c^5 e^3 x^5 b^2+192512 c^5 d e^2 x^4 b^2-3456 a c^4 e^3 x^3 b^2+247296 c^5 d^2 e x^3 b^2+118272 a^2 c^3 d e^2 b^2+114688 c^5 d^3 x^2 b^2-15360 a c^4 d e^2 x^2 b^2+19104 a^2 c^3 e^3 x b^2-21504 a c^4 d^2 e x b^2+128000 c^6 e^3 x^6 b+450560 c^6 d e^2 x^5 b+72192 a c^5 e^3 x^4 b+544768 c^6 d^2 e x^4 b+42432 a^3 c^3 e^3 b+229376 c^6 d^3 x^3 b+284672 a c^5 d e^2 x^3 b-11136 a^2 c^4 e^3 x^2 b+408576 a c^5 d^2 e x^2 b-118272 a^2 c^4 d^2 e b+229376 a c^5 d^3 x b-58368 a^2 c^4 d e^2 x b+71680 c^7 e^3 x^7+245760 c^7 d e^2 x^6+107520 a c^6 e^3 x^5+286720 c^7 d^2 e x^5+114688 c^7 d^3 x^4+393216 a c^6 d e^2 x^4+114688 a^2 c^5 d^3+8960 a^2 c^5 e^3 x^3+501760 a c^6 d^2 e x^3-98304 a^3 c^4 d e^2+229376 a c^6 d^3 x^2+49152 a^2 c^5 d e^2 x^2-13440 a^3 c^4 e^3 x+107520 a^2 c^5 d^2 e x\right )}{286720 c^5}-\frac {3 \left (9 e^3 b^8-32 c d e^2 b^7-112 a c e^3 b^6+32 c^2 d^2 e b^6+384 a c^2 d e^2 b^5+480 a^2 c^2 e^3 b^4-384 a c^3 d^2 e b^4-1536 a^2 c^3 d e^2 b^3-768 a^3 c^3 e^3 b^2+1536 a^2 c^4 d^2 e b^2+2048 a^3 c^4 d e^2 b+256 a^4 c^4 e^3-2048 a^3 c^5 d^2 e\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{16384 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 1587, normalized size = 4.19
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 856, normalized size = 2.26 \begin {gather*} \frac {1}{286720} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, c^{2} x e^{3} + \frac {48 \, c^{9} d e^{2} + 25 \, b c^{8} e^{3}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} e + 352 \, b c^{8} d e^{2} + 41 \, b^{2} c^{7} e^{3} + 84 \, a c^{8} e^{3}}{c^{7}}\right )} x + \frac {896 \, c^{9} d^{3} + 4256 \, b c^{8} d^{2} e + 1504 \, b^{2} c^{7} d e^{2} + 3072 \, a c^{8} d e^{2} - 3 \, b^{3} c^{6} e^{3} + 564 \, a b c^{7} e^{3}}{c^{7}}\right )} x + \frac {14336 \, b c^{8} d^{3} + 15456 \, b^{2} c^{7} d^{2} e + 31360 \, a c^{8} d^{2} e - 96 \, b^{3} c^{6} d e^{2} + 17792 \, a b c^{7} d e^{2} + 27 \, b^{4} c^{5} e^{3} - 216 \, a b^{2} c^{6} e^{3} + 560 \, a^{2} c^{7} e^{3}}{c^{7}}\right )} x + \frac {14336 \, b^{2} c^{7} d^{3} + 28672 \, a c^{8} d^{3} - 224 \, b^{3} c^{6} d^{2} e + 51072 \, a b c^{7} d^{2} e + 224 \, b^{4} c^{5} d e^{2} - 1920 \, a b^{2} c^{6} d e^{2} + 6144 \, a^{2} c^{7} d e^{2} - 63 \, b^{5} c^{4} e^{3} + 568 \, a b^{3} c^{5} e^{3} - 1392 \, a^{2} b c^{6} e^{3}}{c^{7}}\right )} x + \frac {114688 \, a b c^{7} d^{3} + 1120 \, b^{4} c^{5} d^{2} e - 10752 \, a b^{2} c^{6} d^{2} e + 53760 \, a^{2} c^{7} d^{2} e - 1120 \, b^{5} c^{4} d e^{2} + 10752 \, a b^{3} c^{5} d e^{2} - 29184 \, a^{2} b c^{6} d e^{2} + 315 \, b^{6} c^{3} e^{3} - 3164 \, a b^{4} c^{4} e^{3} + 9552 \, a^{2} b^{2} c^{5} e^{3} - 6720 \, a^{3} c^{6} e^{3}}{c^{7}}\right )} x + \frac {114688 \, a^{2} c^{7} d^{3} - 3360 \, b^{5} c^{4} d^{2} e + 35840 \, a b^{3} c^{5} d^{2} e - 118272 \, a^{2} b c^{6} d^{2} e + 3360 \, b^{6} c^{3} d e^{2} - 35840 \, a b^{4} c^{4} d e^{2} + 118272 \, a^{2} b^{2} c^{5} d e^{2} - 98304 \, a^{3} c^{6} d e^{2} - 945 \, b^{7} c^{2} e^{3} + 10500 \, a b^{5} c^{3} e^{3} - 37744 \, a^{2} b^{3} c^{4} e^{3} + 42432 \, a^{3} b c^{5} e^{3}}{c^{7}}\right )} - \frac {3 \, {\left (32 \, b^{6} c^{2} d^{2} e - 384 \, a b^{4} c^{3} d^{2} e + 1536 \, a^{2} b^{2} c^{4} d^{2} e - 2048 \, a^{3} c^{5} d^{2} e - 32 \, b^{7} c d e^{2} + 384 \, a b^{5} c^{2} d e^{2} - 1536 \, a^{2} b^{3} c^{3} d e^{2} + 2048 \, a^{3} b c^{4} d e^{2} + 9 \, b^{8} e^{3} - 112 \, a b^{6} c e^{3} + 480 \, a^{2} b^{4} c^{2} e^{3} - 768 \, a^{3} b^{2} c^{3} e^{3} + 256 \, a^{4} c^{4} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16384 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1607, normalized size = 4.24
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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