3.30.36 \(\int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx\) [2936]

3.30.36.1 Optimal result

Integrand size = 21, antiderivative size = 191 \[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\frac {2 a^4 x^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^5 \left (c x^2\right )^{5/2}}-\frac {8 a^3 x^5 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^5 \left (c x^2\right )^{5/2}}+\frac {12 a^2 x^5 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^5 \left (c x^2\right )^{5/2}}-\frac {8 a x^5 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^5 \left (c x^2\right )^{5/2}}+\frac {2 x^5 \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^5 \left (c x^2\right )^{5/2}} \]

output

2/3*a^4*x^5*(a+b*(c*x^2)^(1/2))^(3/2)/b^5/(c*x^2)^(5/2)-8/5*a^3*x^5*(a+b*( 
c*x^2)^(1/2))^(5/2)/b^5/(c*x^2)^(5/2)+12/7*a^2*x^5*(a+b*(c*x^2)^(1/2))^(7/ 
2)/b^5/(c*x^2)^(5/2)-8/9*a*x^5*(a+b*(c*x^2)^(1/2))^(9/2)/b^5/(c*x^2)^(5/2) 
+2/11*x^5*(a+b*(c*x^2)^(1/2))^(11/2)/b^5/(c*x^2)^(5/2)
 

3.30.36.2 Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.50 \[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\frac {2 x \left (a+b \sqrt {c x^2}\right )^{3/2} \left (128 a^4+240 a^2 b^2 c x^2+315 b^4 c^2 x^4-192 a^3 b \sqrt {c x^2}-280 a b^3 \left (c x^2\right )^{3/2}\right )}{3465 b^5 c^2 \sqrt {c x^2}} \]

input

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

output

(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])
 

3.30.36.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {892, 53, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 892

\(\displaystyle \frac {x^5 \int c^2 x^4 \sqrt {a+b \sqrt {c x^2}}d\sqrt {c x^2}}{\left (c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {x^5 \int \left (\frac {\left (a+b \sqrt {c x^2}\right )^{9/2}}{b^4}-\frac {4 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{b^4}+\frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^4}-\frac {4 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{b^4}+\frac {a^4 \sqrt {a+b \sqrt {c x^2}}}{b^4}\right )d\sqrt {c x^2}}{\left (c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^5 \left (\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^5}-\frac {8 a^3 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^5}+\frac {12 a^2 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^5}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^5}-\frac {8 a \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^5}\right )}{\left (c x^2\right )^{5/2}}\)

input

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

output

(x^5*((2*a^4*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^5) - (8*a^3*(a + b*Sqrt[c*x^2 
])^(5/2))/(5*b^5) + (12*a^2*(a + b*Sqrt[c*x^2])^(7/2))/(7*b^5) - (8*a*(a + 
 b*Sqrt[c*x^2])^(9/2))/(9*b^5) + (2*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^5))) 
/(c*x^2)^(5/2)
 

3.30.36.3.1 Defintions of rubi rules used

rule 53

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] && 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])
 

rule 892

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> Simp[(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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.30.36.4 Maple [A] (verified)

Time = 3.83 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.44

method	result	size
default	\(-\frac {2 x^{5} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}} \left (-315 c^{2} x^{4} b^{4}+280 \left (c \,x^{2}\right )^{\frac {3}{2}} a \,b^{3}-240 c \,x^{2} a^{2} b^{2}+192 \sqrt {c \,x^{2}}\, a^{3} b -128 a^{4}\right )}{3465 \left (c \,x^{2}\right )^{\frac {5}{2}} b^{5}}\)	\(84\)

input

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

output

-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
 

3.30.36.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\frac {2 \, {\left (315 \, b^{5} c^{3} x^{6} - 40 \, a^{2} b^{3} c^{2} x^{4} - 64 \, a^{4} b c x^{2} + {\left (35 \, a b^{4} c^{2} x^{4} + 48 \, a^{3} b^{2} c x^{2} + 128 \, a^{5}\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{3465 \, b^{5} c^{3} x} \]

input

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

output

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)
 

3.30.36.6 Sympy [F]

\[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\int x^{4} \sqrt {a + b \sqrt {c x^{2}}}\, dx \]

input

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

output

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

3.30.36.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2192 vs. \(2 (151) = 302\).

Time = 0.47 (sec) , antiderivative size = 2192, normalized size of antiderivative = 11.48 \[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\text {Too large to display} \]

input

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

output

((253*c^33 + 2558956*c^32 + 7217549950*c^31 + 8987703765844*c^30 + 6036468 
373437617*c^29 + 2446429529849811272*c^28 + 642455910258816305144*c^27 + 1 
14777366281226527056208*c^26 + 14444206931227366367330858*c^25 + 131365425 
6537258900978878920*c^24 + 88007787535651613090646185140*c^23 + 4405711003 
982878865632262198872*c^22 + 166544000020720524719921573991514*c^21 + 4789 
438716064434805459841864162048*c^20 + 105284116366548048830595983583302024 
*c^19 + 1773444928146150427905082217087812880*c^18 + 228948392597758710018 
29906064713305625*c^17 + 226076660023411473110523953150238987500*c^16 + 17 
00246465246927686150050738273824218750*c^15 + 9672993246548251837557896244 
481445312500*c^14 + 41230185720792035261437425937884033203125*c^13 + 12995 
6781520382049850939376902099609375000*c^12 + 29769407278591668426367728464 
1113281250000*c^11 + 484329529502415188750357304687500000000000*c^10 + 542 
693518652974490238804687500000000000000*c^9 + 4015597375339545508887500000 
00000000000000*c^8 + 184849853908622316875000000000000000000000*c^7 + 4839 
4254985190280000000000000000000000000*c^6 + 621216412668000000000000000000 
0000000000*c^5 + 292206528000000000000000000000000000000*c^4 + 20995200000 
00000000000000000000000000*c^3 + (c^33 + 31444*c^32 + 153361414*c^31 + 277 
761034468*c^30 + 249531421449205*c^29 + 128781547874762192*c^28 + 41710765 
820505500216*c^27 + 8988868827121079441936*c^26 + 134284078049474894776670 
6*c^25 + 143266166424564257427917848*c^24 + 111599953405280930042187806...
 

3.30.36.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.05 \[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\frac {2 \, {\left (\frac {11 \, {\left (35 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sqrt {c} x + a} a^{4}\right )} a}{b^{4} c^{2}} + \frac {5 \, {\left (63 \, {\left (b \sqrt {c} x + a\right )}^{\frac {11}{2}} \sqrt {c} - 385 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} a \sqrt {c} + 990 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a^{2} \sqrt {c} - 1386 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{3} \sqrt {c} + 1155 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{4} \sqrt {c} - 693 \, \sqrt {b \sqrt {c} x + a} a^{5} \sqrt {c}\right )}}{b^{4} c^{\frac {5}{2}}}\right )}}{3465 \, b \sqrt {c}} \]

input

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

output

2/3465*(11*(35*(b*sqrt(c)*x + a)^(9/2) - 180*(b*sqrt(c)*x + a)^(7/2)*a + 3 
78*(b*sqrt(c)*x + a)^(5/2)*a^2 - 420*(b*sqrt(c)*x + a)^(3/2)*a^3 + 315*sqr 
t(b*sqrt(c)*x + a)*a^4)*a/(b^4*c^2) + 5*(63*(b*sqrt(c)*x + a)^(11/2)*sqrt( 
c) - 385*(b*sqrt(c)*x + a)^(9/2)*a*sqrt(c) + 990*(b*sqrt(c)*x + a)^(7/2)*a 
^2*sqrt(c) - 1386*(b*sqrt(c)*x + a)^(5/2)*a^3*sqrt(c) + 1155*(b*sqrt(c)*x 
+ a)^(3/2)*a^4*sqrt(c) - 693*sqrt(b*sqrt(c)*x + a)*a^5*sqrt(c))/(b^4*c^(5/ 
2)))/(b*sqrt(c))
 

3.30.36.9 Mupad [F(-1)]

Timed out. \[ \int x^4 \sqrt {a+b \sqrt {c x^2}} \, dx=\int x^4\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \]

input

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

output

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