∫ x4∘ ______ a + b√ _

3.2936 \(\int x^4 \sqrt{a+b \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{12 a^2 x^5 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a^3 x^5 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^5 \left (c x^2\right )^{5/2}}+\frac{2 a^4 x^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^5 \left (c x^2\right )^{5/2}}+\frac{2 x^5 \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a x^5 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^5 \left (c x^2\right )^{5/2}} \]

[Out]

(2*a^4*x^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^5*(c*x^2)^(5/2)) - (8*a^3*x^5*(a + b*Sqrt[c*x^2])^(5/2))/(5*b^5*(c*

x^2)^(5/2)) + (12*a^2*x^5*(a + b*Sqrt[c*x^2])^(7/2))/(7*b^5*(c*x^2)^(5/2)) - (8*a*x^5*(a + b*Sqrt[c*x^2])^(9/2

))/(9*b^5*(c*x^2)^(5/2)) + (2*x^5*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^5*(c*x^2)^(5/2))

________________________________________________________________________________________

Rubi [A] time = 0.0680984, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {368, 43} \[ \frac{12 a^2 x^5 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a^3 x^5 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^5 \left (c x^2\right )^{5/2}}+\frac{2 a^4 x^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^5 \left (c x^2\right )^{5/2}}+\frac{2 x^5 \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a x^5 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^5 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[x^4*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(2*a^4*x^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^5*(c*x^2)^(5/2)) - (8*a^3*x^5*(a + b*Sqrt[c*x^2])^(5/2))/(5*b^5*(c*

x^2)^(5/2)) + (12*a^2*x^5*(a + b*Sqrt[c*x^2])^(7/2))/(7*b^5*(c*x^2)^(5/2)) - (8*a*x^5*(a + b*Sqrt[c*x^2])^(9/2

))/(9*b^5*(c*x^2)^(5/2)) + (2*x^5*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^5*(c*x^2)^(5/2))

Rule 368

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q

)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q

}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d

*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^4 \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{x^5 \operatorname{Subst}\left (\int x^4 \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{5/2}}\\ &=\frac{x^5 \operatorname{Subst}\left (\int \left (\frac{a^4 \sqrt{a+b x}}{b^4}-\frac{4 a^3 (a+b x)^{3/2}}{b^4}+\frac{6 a^2 (a+b x)^{5/2}}{b^4}-\frac{4 a (a+b x)^{7/2}}{b^4}+\frac{(a+b x)^{9/2}}{b^4}\right ) \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{5/2}}\\ &=\frac{2 a^4 x^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a^3 x^5 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^5 \left (c x^2\right )^{5/2}}+\frac{12 a^2 x^5 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a x^5 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^5 \left (c x^2\right )^{5/2}}+\frac{2 x^5 \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^5 \left (c x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A] time = 0.0478401, size = 96, normalized size = 0.5 \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2} \left (240 a^2 b^2 c x^2-192 a^3 b \sqrt{c x^2}+128 a^4-280 a b^3 \left (c x^2\right )^{3/2}+315 b^4 c^2 x^4\right )}{3465 b^5 c^2 \sqrt{c x^2}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^4*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(2*x*(a + b*Sqrt[c*x^2])^(3/2)*(128*a^4 + 240*a^2*b^2*c*x^2 + 315*b^4*c^2*x^4 - 192*a^3*b*Sqrt[c*x^2] - 280*a*

b^3*(c*x^2)^(3/2)))/(3465*b^5*c^2*Sqrt[c*x^2])

________________________________________________________________________________________

Maple [A] time = 0.007, size = 84, normalized size = 0.4 \begin{align*} -{\frac{2\,{x}^{5}}{3465\,{b}^{5}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( -315\,{c}^{2}{x}^{4}{b}^{4}+280\, \left ( c{x}^{2} \right ) ^{3/2}a{b}^{3}-240\,c{x}^{2}{a}^{2}{b}^{2}+192\,\sqrt{c{x}^{2}}{a}^{3}b-128\,{a}^{4} \right ) \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*(c*x^2)^(1/2))^(1/2),x)

[Out]

-2/3465*x^5*(a+b*(c*x^2)^(1/2))^(3/2)*(-315*c^2*x^4*b^4+280*(c*x^2)^(3/2)*a*b^3-240*c*x^2*a^2*b^2+192*(c*x^2)^

(1/2)*a^3*b-128*a^4)/(c*x^2)^(5/2)/b^5

________________________________________________________________________________________

Maxima [B] time = 1.50976, size = 2959, normalized size = 15.49 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

((253*c^33 + 2558956*c^32 + 7217549950*c^31 + 8987703765844*c^30 + 6036468373437617*c^29 + 2446429529849811272

*c^28 + 642455910258816305144*c^27 + 114777366281226527056208*c^26 + 14444206931227366367330858*c^25 + 1313654

256537258900978878920*c^24 + 88007787535651613090646185140*c^23 + 4405711003982878865632262198872*c^22 + 16654

4000020720524719921573991514*c^21 + 4789438716064434805459841864162048*c^20 + 10528411636654804883059598358330

2024*c^19 + 1773444928146150427905082217087812880*c^18 + 22894839259775871001829906064713305625*c^17 + 2260766

60023411473110523953150238987500*c^16 + 1700246465246927686150050738273824218750*c^15 + 9672993246548251837557

896244481445312500*c^14 + 41230185720792035261437425937884033203125*c^13 + 12995678152038204985093937690209960

9375000*c^12 + 297694072785916684263677284641113281250000*c^11 + 484329529502415188750357304687500000000000*c^

10 + 542693518652974490238804687500000000000000*c^9 + 401559737533954550888750000000000000000000*c^8 + 1848498

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8 + 41710765820505500216*c^27 + 8988868827121079441936*c^26 + 1342840780494748947766706*c^25 + 143266166424564

257427917848*c^24 + 11159995340528093004218780644*c^23 + 645308253745684073542574566840*c^22 + 280407759850682

92094392289155714*c^21 + 923839137257607053430155106920656*c^20 + 23217713909073497257593845317682600*c^19 + 4

46699434346765451578586034218289616*c^18 + 6587962278317998652016627783149754125*c^17 + 7441133104745009071492

6280659005912500*c^16 + 641687702611309496545321307620461343750*c^15 + 420162704040226895719531612962660156250

0*c^14 + 20719469703534769824847871522785791015625*c^13 + 76087449461581872206557314710546875000000*c^12 + 204

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671682730487700000*c^16 + 260004289926401703985972418707308593750*c^15 + 1560709818236985476370671875848242187

500*c^14 + 6998853809133530096547036558431884765625*c^13 + 23148367524550212705560195893920898437500*c^12 + 55

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288187500*c^15 + 660229305541320870206161063444589843750*c^14 + 3430153973600309932075208741088476562500*c^13

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6155422385182353771972656250000*c^10 + 107061119041927298564775805664062500000000*c^9 + 1021227325957172475146

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46*c^29 + 195743173114099*c^28 + 94380747307709584*c^27 + 28462469351818276960*c^26 + 5690177551338578052424*c

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4403980255*c^27 + 85951284212292484*c^26 + 25344638059811815968*c^25 + 4946658491131916154584*c^24 + 665454644

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90099487304687500*c^10 + 7055588854027909397802579956054687500000*c^9 + 93364705099387213396354980468750000000

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00000000000000*c^5 + 413670210404287500000000000000000000000*c^4 + 44125007175000000000000000000000000000*c^3

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*c^16 + 17623315230142149140532220845272500*c^15 + 242505343547381820211816815297262500*c^14 + 251055864929126

6783281237190873046875*c^13 + 19454943027839870676475341947998046875*c^12 + 1119312615303221509900054695988769

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69989949087038610662*c^21 + 57366549664221202749288378*c^20 + 3905991055707128875010770926*c^19 + 195602162590

058562386788777038*c^18 + 7285323827632700990474618061840*c^17 + 203358338393512800465448614893484*c^16 + 4273

475844388426205809239967486500*c^15 + 67727208633048929025206537495962500*c^14 + 80872735891503824805019801633

5140625*c^13 + 7249566807948818893161180513779296875*c^12 + 48466044976177871402659958222509765625*c^11 + 2392

66096150056760942527466119384765625*c^10 + 860235682244821957555057436828613281250*c^9 + 221025113708682441371

3384136962890625000*c^8 + 3955969049479281072819201660156250000000*c^7 + 4763019763214843448465820312500000000

000*c^6 + 3673406927808397595703125000000000000000*c^5 + 1689153827145141718750000000000000000000*c^4 + 413670

210404287500000000000000000000000*c^3 + 44125007175000000000000000000000000000*c^2 + (243*c^28 + 2260611*c^27

+ 5839685133*c^26 + 6629674620105*c^25 + 4039306818348400*c^24 + 1477003668609286412*c^23 + 347891895712128419

364*c^22 + 55384654169497405674580*c^21 + 6166756400536006313788202*c^20 + 492319554678764362836685050*c^19 +

28700362927999647352259163382*c^18 + 1238061453504658795976684722622*c^17 + 39889694031955537300197769308276*c

^16 + 965736374745116134671262453002620*c^15 + 17623315230142149140532220845272500*c^14 + 24250534354738182021

1816815297262500*c^13 + 2510558649291266783281237190873046875*c^12 + 19454943027839870676475341947998046875*c^

11 + 111931261530322150990005469598876953125*c^10 + 472517250009358477114024029449462890625*c^9 + 144007006378

5699316009505962524414062500*c^8 + 3099619804548628324983378295898437500000*c^7 + 4573450746723877891169677734

375000000000*c^6 + 4439227951799894634199218750000000000000*c^5 + 2672664979778960273437500000000000000000*c^4

+ 913107736622173125000000000000000000000*c^3 + 152730150851250000000000000000000000000*c^2 + 935343450000000

0000000000000000000000*c + 87480000000000000000000000000000000)*sqrt(c) + 129324600000000000000000000000000000

0*c)*a^5)*sqrt(b*sqrt(c)*x + a)/((c^34 + 32709*c^33 + 166156194*c^32 + 313848784218*c^31 + 294469940278425*c^3

0 + 158963889741950277*c^29 + 53942913469754556576*c^28 + 12201148378415160967656*c^27 + 191672761190088158304

7746*c^26 + 215487201080701089264572138*c^25 + 17728266623214387509113175244*c^24 + 10853471914239421389958054

92540*c^23 + 50069331004982686422553600150074*c^22 + 1756559137361209677029762976878226*c^21 + 471649074893956

71284893054638492840*c^20 + 973120016179505695731565952134799736*c^19 + 15455186919048750791542038868588818525

*c^18 + 188885527346329445724075810982572440625*c^17 + 1772071002728366862097941073371656281250*c^16 + 1270285

9366636907387945569820995722656250*c^15 + 69084435936276029012637352745193017578125*c^14 + 2822383780655420485

13744444399967041015625*c^13 + 854744024058075360839114990533447265625000*c^12 + 18854538254093270834055112130

49316406250000*c^11 + 2960034820194099448691930273437500000000000*c^10 + 3206726511270252085969898437500000000

000000*c^9 + 2298258834452828289318750000000000000000000*c^8 + 1026494232105541204375000000000000000000000*c^7

+ 261175070360341800000000000000000000000000*c^6 + 32633630940600000000000000000000000000000*c^5 + 1496444544

000000000000000000000000000000*c^4 + 10497600000000000000000000000000000000*c^3 + 2*(129*c^33 + 1358088*c^32 +

3992178510*c^31 + 5188254469092*c^30 + 3642062740341821*c^29 + 1545168634611811116*c^28 + 4255048696806719031

12*c^27 + 79860855208415962132944*c^26 + 10579205416850555553082194*c^25 + 1014992544330040094059234080*c^24 +

71903882119146039055870044180*c^23 + 3816126136355649616672567516536*c^22 + 153373939973030992595941509885042

*c^21 + 4704317201176235036305308699382664*c^20 + 110686342955957767559282605085857512*c^19 + 2003471049939988

842899006194089630480*c^18 + 27917325325682932130956522490231038125*c^17 + 29906665763033096334257767822263427

5000*c^16 + 2454342489151737584438328638188065468750*c^15 + 15340564224279798311767238446307226562500*c^14 + 7

2413767119232942192838391775906494140625*c^13 + 255197014414145705441862975227416992187500*c^12 + 661247327533

371121092883907377929687500000*c^11 + 1234623418450566749592990626953125000000000*c^10 + 161731469103154600746

9761718750000000000000*c^9 + 1433927163780426362384062500000000000000000*c^8 + 8185752939119499956250000000000

00000000000*c^7 + 279809533898669190000000000000000000000000*c^6 + 51115570649316000000000000000000000000000*c

^5 + 4078129032000000000000000000000000000000*c^4 + 89579520000000000000000000000000000000*c^3)*sqrt(c))*b^5)

________________________________________________________________________________________

Fricas [A] time = 1.26342, size = 219, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (315 \, b^{5} c^{3} x^{6} - 40 \, a^{2} b^{3} c^{2} x^{4} - 64 \, a^{4} b c x^{2} +{\left (35 \, a b^{4} c^{2} x^{4} + 48 \, a^{3} b^{2} c x^{2} + 128 \, a^{5}\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3465 \, b^{5} c^{3} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*b^5*c^3*x^6 - 40*a^2*b^3*c^2*x^4 - 64*a^4*b*c*x^2 + (35*a*b^4*c^2*x^4 + 48*a^3*b^2*c*x^2 + 128*a^5

)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a)/(b^5*c^3*x)

________________________________________________________________________________________

Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*(c*x**2)**(1/2))**(1/2),x)

[Out]

Integral(x**4*sqrt(a + b*sqrt(c*x**2)), x)

________________________________________________________________________________________

Giac [A] time = 1.17681, size = 107, normalized size = 0.56 \begin{align*} \frac{2 \,{\left (315 \,{\left (b \sqrt{c} x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{4}\right )}}{3465 \, b^{5} c^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

2/3465*(315*(b*sqrt(c)*x + a)^(11/2) - 1540*(b*sqrt(c)*x + a)^(9/2)*a + 2970*(b*sqrt(c)*x + a)^(7/2)*a^2 - 277

2*(b*sqrt(c)*x + a)^(5/2)*a^3 + 1155*(b*sqrt(c)*x + a)^(3/2)*a^4)/(b^5*c^(5/2))

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