\(\int (a+b \sqrt {x})^p x^3 \, dx\) [154]

Optimal result

Integrand size = 15, antiderivative size = 204 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=-\frac {2 a^7 \left (a+b \sqrt {x}\right )^{1+p}}{b^8 (1+p)}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{2+p}}{b^8 (2+p)}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{3+p}}{b^8 (3+p)}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{4+p}}{b^8 (4+p)}-\frac {70 a^3 \left (a+b \sqrt {x}\right )^{5+p}}{b^8 (5+p)}+\frac {42 a^2 \left (a+b \sqrt {x}\right )^{6+p}}{b^8 (6+p)}-\frac {14 a \left (a+b \sqrt {x}\right )^{7+p}}{b^8 (7+p)}+\frac {2 \left (a+b \sqrt {x}\right )^{8+p}}{b^8 (8+p)} \] Output:

-2*a^7*(a+b*x^(1/2))^(p+1)/b^8/(p+1)+14*a^6*(a+b*x^(1/2))^(2+p)/b^8/(2+p)- 
42*a^5*(a+b*x^(1/2))^(3+p)/b^8/(3+p)+70*a^4*(a+b*x^(1/2))^(4+p)/b^8/(4+p)- 
70*a^3*(a+b*x^(1/2))^(5+p)/b^8/(5+p)+42*a^2*(a+b*x^(1/2))^(6+p)/b^8/(6+p)- 
14*a*(a+b*x^(1/2))^(7+p)/b^8/(7+p)+2*(a+b*x^(1/2))^(8+p)/b^8/(8+p)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.82 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=\frac {2 \left (-\frac {a^7}{1+p}+\frac {7 a^6 \left (a+b \sqrt {x}\right )}{2+p}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^2}{3+p}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^3}{4+p}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^4}{5+p}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^5}{6+p}-\frac {7 a \left (a+b \sqrt {x}\right )^6}{7+p}+\frac {\left (a+b \sqrt {x}\right )^7}{8+p}\right ) \left (a+b \sqrt {x}\right )^{1+p}}{b^8} \] Input:

Integrate[(a + b*Sqrt[x])^p*x^3,x]
 

Output:

(2*(-(a^7/(1 + p)) + (7*a^6*(a + b*Sqrt[x]))/(2 + p) - (21*a^5*(a + b*Sqrt 
[x])^2)/(3 + p) + (35*a^4*(a + b*Sqrt[x])^3)/(4 + p) - (35*a^3*(a + b*Sqrt 
[x])^4)/(5 + p) + (21*a^2*(a + b*Sqrt[x])^5)/(6 + p) - (7*a*(a + b*Sqrt[x] 
)^6)/(7 + p) + (a + b*Sqrt[x])^7/(8 + p))*(a + b*Sqrt[x])^(1 + p))/b^8
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 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.

\(\displaystyle \int x^3 \left (a+b \sqrt {x}\right )^p \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^p x^{7/2}d\sqrt {x}\)

\(\Big \downarrow \) 53

\(\displaystyle 2 \int \left (-\frac {a^7 \left (a+b \sqrt {x}\right )^p}{b^7}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{p+1}}{b^7}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^{p+2}}{b^7}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^{p+3}}{b^7}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^{p+4}}{b^7}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{p+5}}{b^7}-\frac {7 a \left (a+b \sqrt {x}\right )^{p+6}}{b^7}+\frac {\left (a+b \sqrt {x}\right )^{p+7}}{b^7}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^7 \left (a+b \sqrt {x}\right )^{p+1}}{b^8 (p+1)}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{p+2}}{b^8 (p+2)}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^{p+3}}{b^8 (p+3)}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^{p+4}}{b^8 (p+4)}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^{p+5}}{b^8 (p+5)}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{p+6}}{b^8 (p+6)}-\frac {7 a \left (a+b \sqrt {x}\right )^{p+7}}{b^8 (p+7)}+\frac {\left (a+b \sqrt {x}\right )^{p+8}}{b^8 (p+8)}\right )\)

Input:

Int[(a + b*Sqrt[x])^p*x^3,x]
 

Output:

2*(-((a^7*(a + b*Sqrt[x])^(1 + p))/(b^8*(1 + p))) + (7*a^6*(a + b*Sqrt[x]) 
^(2 + p))/(b^8*(2 + p)) - (21*a^5*(a + b*Sqrt[x])^(3 + p))/(b^8*(3 + p)) + 
 (35*a^4*(a + b*Sqrt[x])^(4 + p))/(b^8*(4 + p)) - (35*a^3*(a + b*Sqrt[x])^ 
(5 + p))/(b^8*(5 + p)) + (21*a^2*(a + b*Sqrt[x])^(6 + p))/(b^8*(6 + p)) - 
(7*a*(a + b*Sqrt[x])^(7 + p))/(b^8*(7 + p)) + (a + b*Sqrt[x])^(8 + p)/(b^8 
*(8 + p)))
 

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 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

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

Maple [F]

\[\int \left (a +b \sqrt {x}\right )^{p} x^{3}d x\]

Input:

int((a+b*x^(1/2))^p*x^3,x)
 

Output:

int((a+b*x^(1/2))^p*x^3,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (188) = 376\).

Time = 0.13 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.25 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=-\frac {2 \, {\left (5040 \, a^{8} - {\left (b^{8} p^{7} + 28 \, b^{8} p^{6} + 322 \, b^{8} p^{5} + 1960 \, b^{8} p^{4} + 6769 \, b^{8} p^{3} + 13132 \, b^{8} p^{2} + 13068 \, b^{8} p + 5040 \, b^{8}\right )} x^{4} + 7 \, {\left (a^{2} b^{6} p^{6} + 15 \, a^{2} b^{6} p^{5} + 85 \, a^{2} b^{6} p^{4} + 225 \, a^{2} b^{6} p^{3} + 274 \, a^{2} b^{6} p^{2} + 120 \, a^{2} b^{6} p\right )} x^{3} + 210 \, {\left (a^{4} b^{4} p^{4} + 6 \, a^{4} b^{4} p^{3} + 11 \, a^{4} b^{4} p^{2} + 6 \, a^{4} b^{4} p\right )} x^{2} + 2520 \, {\left (a^{6} b^{2} p^{2} + a^{6} b^{2} p\right )} x - {\left (5040 \, a^{7} b p + {\left (a b^{7} p^{7} + 21 \, a b^{7} p^{6} + 175 \, a b^{7} p^{5} + 735 \, a b^{7} p^{4} + 1624 \, a b^{7} p^{3} + 1764 \, a b^{7} p^{2} + 720 \, a b^{7} p\right )} x^{3} + 42 \, {\left (a^{3} b^{5} p^{5} + 10 \, a^{3} b^{5} p^{4} + 35 \, a^{3} b^{5} p^{3} + 50 \, a^{3} b^{5} p^{2} + 24 \, a^{3} b^{5} p\right )} x^{2} + 840 \, {\left (a^{5} b^{3} p^{3} + 3 \, a^{5} b^{3} p^{2} + 2 \, a^{5} b^{3} p\right )} x\right )} \sqrt {x}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{b^{8} p^{8} + 36 \, b^{8} p^{7} + 546 \, b^{8} p^{6} + 4536 \, b^{8} p^{5} + 22449 \, b^{8} p^{4} + 67284 \, b^{8} p^{3} + 118124 \, b^{8} p^{2} + 109584 \, b^{8} p + 40320 \, b^{8}} \] Input:

integrate((a+b*x^(1/2))^p*x^3,x, algorithm="fricas")
 

Output:

-2*(5040*a^8 - (b^8*p^7 + 28*b^8*p^6 + 322*b^8*p^5 + 1960*b^8*p^4 + 6769*b 
^8*p^3 + 13132*b^8*p^2 + 13068*b^8*p + 5040*b^8)*x^4 + 7*(a^2*b^6*p^6 + 15 
*a^2*b^6*p^5 + 85*a^2*b^6*p^4 + 225*a^2*b^6*p^3 + 274*a^2*b^6*p^2 + 120*a^ 
2*b^6*p)*x^3 + 210*(a^4*b^4*p^4 + 6*a^4*b^4*p^3 + 11*a^4*b^4*p^2 + 6*a^4*b 
^4*p)*x^2 + 2520*(a^6*b^2*p^2 + a^6*b^2*p)*x - (5040*a^7*b*p + (a*b^7*p^7 
+ 21*a*b^7*p^6 + 175*a*b^7*p^5 + 735*a*b^7*p^4 + 1624*a*b^7*p^3 + 1764*a*b 
^7*p^2 + 720*a*b^7*p)*x^3 + 42*(a^3*b^5*p^5 + 10*a^3*b^5*p^4 + 35*a^3*b^5* 
p^3 + 50*a^3*b^5*p^2 + 24*a^3*b^5*p)*x^2 + 840*(a^5*b^3*p^3 + 3*a^5*b^3*p^ 
2 + 2*a^5*b^3*p)*x)*sqrt(x))*(b*sqrt(x) + a)^p/(b^8*p^8 + 36*b^8*p^7 + 546 
*b^8*p^6 + 4536*b^8*p^5 + 22449*b^8*p^4 + 67284*b^8*p^3 + 118124*b^8*p^2 + 
 109584*b^8*p + 40320*b^8)
 

Sympy [F(-2)]

Exception generated. \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((a+b*x**(1/2))**p*x**3,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.40 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=\frac {2 \, {\left ({\left (p^{7} + 28 \, p^{6} + 322 \, p^{5} + 1960 \, p^{4} + 6769 \, p^{3} + 13132 \, p^{2} + 13068 \, p + 5040\right )} b^{8} x^{4} + {\left (p^{7} + 21 \, p^{6} + 175 \, p^{5} + 735 \, p^{4} + 1624 \, p^{3} + 1764 \, p^{2} + 720 \, p\right )} a b^{7} x^{\frac {7}{2}} - 7 \, {\left (p^{6} + 15 \, p^{5} + 85 \, p^{4} + 225 \, p^{3} + 274 \, p^{2} + 120 \, p\right )} a^{2} b^{6} x^{3} + 42 \, {\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} a^{3} b^{5} x^{\frac {5}{2}} - 210 \, {\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} a^{4} b^{4} x^{2} + 840 \, {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a^{5} b^{3} x^{\frac {3}{2}} - 2520 \, {\left (p^{2} + p\right )} a^{6} b^{2} x + 5040 \, a^{7} b p \sqrt {x} - 5040 \, a^{8}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{{\left (p^{8} + 36 \, p^{7} + 546 \, p^{6} + 4536 \, p^{5} + 22449 \, p^{4} + 67284 \, p^{3} + 118124 \, p^{2} + 109584 \, p + 40320\right )} b^{8}} \] Input:

integrate((a+b*x^(1/2))^p*x^3,x, algorithm="maxima")
 

Output:

2*((p^7 + 28*p^6 + 322*p^5 + 1960*p^4 + 6769*p^3 + 13132*p^2 + 13068*p + 5 
040)*b^8*x^4 + (p^7 + 21*p^6 + 175*p^5 + 735*p^4 + 1624*p^3 + 1764*p^2 + 7 
20*p)*a*b^7*x^(7/2) - 7*(p^6 + 15*p^5 + 85*p^4 + 225*p^3 + 274*p^2 + 120*p 
)*a^2*b^6*x^3 + 42*(p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*a^3*b^5*x^(5/2) 
 - 210*(p^4 + 6*p^3 + 11*p^2 + 6*p)*a^4*b^4*x^2 + 840*(p^3 + 3*p^2 + 2*p)* 
a^5*b^3*x^(3/2) - 2520*(p^2 + p)*a^6*b^2*x + 5040*a^7*b*p*sqrt(x) - 5040*a 
^8)*(b*sqrt(x) + a)^p/((p^8 + 36*p^7 + 546*p^6 + 4536*p^5 + 22449*p^4 + 67 
284*p^3 + 118124*p^2 + 109584*p + 40320)*b^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (188) = 376\).

Time = 0.13 (sec) , antiderivative size = 1642, normalized size of antiderivative = 8.05 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=\text {Too large to display} \] Input:

integrate((a+b*x^(1/2))^p*x^3,x, algorithm="giac")
 

Output:

2*((b*sqrt(x) + a)^8*(b*sqrt(x) + a)^p*p^7 - 7*(b*sqrt(x) + a)^7*(b*sqrt(x 
) + a)^p*a*p^7 + 21*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*a^2*p^7 - 35*(b*sq 
rt(x) + a)^5*(b*sqrt(x) + a)^p*a^3*p^7 + 35*(b*sqrt(x) + a)^4*(b*sqrt(x) + 
 a)^p*a^4*p^7 - 21*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^5*p^7 + 7*(b*sqrt 
(x) + a)^2*(b*sqrt(x) + a)^p*a^6*p^7 - (b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a 
^7*p^7 + 28*(b*sqrt(x) + a)^8*(b*sqrt(x) + a)^p*p^6 - 203*(b*sqrt(x) + a)^ 
7*(b*sqrt(x) + a)^p*a*p^6 + 630*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*a^2*p^ 
6 - 1085*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a^3*p^6 + 1120*(b*sqrt(x) + a 
)^4*(b*sqrt(x) + a)^p*a^4*p^6 - 693*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^ 
5*p^6 + 238*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^6*p^6 - 35*(b*sqrt(x) + 
a)*(b*sqrt(x) + a)^p*a^7*p^6 + 322*(b*sqrt(x) + a)^8*(b*sqrt(x) + a)^p*p^5 
 - 2401*(b*sqrt(x) + a)^7*(b*sqrt(x) + a)^p*a*p^5 + 7686*(b*sqrt(x) + a)^6 
*(b*sqrt(x) + a)^p*a^2*p^5 - 13685*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a^3 
*p^5 + 14630*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^4*p^5 - 9387*(b*sqrt(x) 
 + a)^3*(b*sqrt(x) + a)^p*a^5*p^5 + 3346*(b*sqrt(x) + a)^2*(b*sqrt(x) + a) 
^p*a^6*p^5 - 511*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^7*p^5 + 1960*(b*sqrt( 
x) + a)^8*(b*sqrt(x) + a)^p*p^4 - 14945*(b*sqrt(x) + a)^7*(b*sqrt(x) + a)^ 
p*a*p^4 + 49140*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*a^2*p^4 - 90335*(b*sqr 
t(x) + a)^5*(b*sqrt(x) + a)^p*a^3*p^4 + 100240*(b*sqrt(x) + a)^4*(b*sqrt(x 
) + a)^p*a^4*p^4 - 67095*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^5*p^4 + ...
 

Mupad [B] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.89 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx={\left (a+b\,\sqrt {x}\right )}^p\,\left (\frac {2\,x^4\,\left (p^7+28\,p^6+322\,p^5+1960\,p^4+6769\,p^3+13132\,p^2+13068\,p+5040\right )}{p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320}-\frac {10080\,a^8}{b^8\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {10080\,a^7\,p\,\sqrt {x}}{b^7\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {5040\,a^6\,p\,x\,\left (p+1\right )}{b^6\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {14\,a^2\,p\,x^3\,\left (p^5+15\,p^4+85\,p^3+225\,p^2+274\,p+120\right )}{b^2\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {1680\,a^5\,p\,x^{3/2}\,\left (p^2+3\,p+2\right )}{b^5\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {420\,a^4\,p\,x^2\,\left (p^3+6\,p^2+11\,p+6\right )}{b^4\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {2\,a\,p\,x^{7/2}\,\left (p^6+21\,p^5+175\,p^4+735\,p^3+1624\,p^2+1764\,p+720\right )}{b\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {84\,a^3\,p\,x^{5/2}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}{b^3\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}\right ) \] Input:

int(x^3*(a + b*x^(1/2))^p,x)
 

Output:

(a + b*x^(1/2))^p*((2*x^4*(13068*p + 13132*p^2 + 6769*p^3 + 1960*p^4 + 322 
*p^5 + 28*p^6 + p^7 + 5040))/(109584*p + 118124*p^2 + 67284*p^3 + 22449*p^ 
4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 + 40320) - (10080*a^8)/(b^8*(109584* 
p + 118124*p^2 + 67284*p^3 + 22449*p^4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 
 + 40320)) + (10080*a^7*p*x^(1/2))/(b^7*(109584*p + 118124*p^2 + 67284*p^3 
 + 22449*p^4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 + 40320)) - (5040*a^6*p*x 
*(p + 1))/(b^6*(109584*p + 118124*p^2 + 67284*p^3 + 22449*p^4 + 4536*p^5 + 
 546*p^6 + 36*p^7 + p^8 + 40320)) - (14*a^2*p*x^3*(274*p + 225*p^2 + 85*p^ 
3 + 15*p^4 + p^5 + 120))/(b^2*(109584*p + 118124*p^2 + 67284*p^3 + 22449*p 
^4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 + 40320)) + (1680*a^5*p*x^(3/2)*(3* 
p + p^2 + 2))/(b^5*(109584*p + 118124*p^2 + 67284*p^3 + 22449*p^4 + 4536*p 
^5 + 546*p^6 + 36*p^7 + p^8 + 40320)) - (420*a^4*p*x^2*(11*p + 6*p^2 + p^3 
 + 6))/(b^4*(109584*p + 118124*p^2 + 67284*p^3 + 22449*p^4 + 4536*p^5 + 54 
6*p^6 + 36*p^7 + p^8 + 40320)) + (2*a*p*x^(7/2)*(1764*p + 1624*p^2 + 735*p 
^3 + 175*p^4 + 21*p^5 + p^6 + 720))/(b*(109584*p + 118124*p^2 + 67284*p^3 
+ 22449*p^4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 + 40320)) + (84*a^3*p*x^(5 
/2)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24))/(b^3*(109584*p + 118124*p^2 + 672 
84*p^3 + 22449*p^4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 + 40320)))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.55 \[ \int \left (a+b \sqrt {x}\right )^p x^3 \, dx=\frac {2 \left (\sqrt {x}\, b +a \right )^{p} \left (b^{8} p^{7} x^{4}+5040 \sqrt {x}\, a^{7} b p -2520 a^{6} b^{2} p^{2} x -2520 a^{6} b^{2} p x -210 a^{4} b^{4} p^{4} x^{2}-1260 a^{4} b^{4} p^{3} x^{2}-2310 a^{4} b^{4} p^{2} x^{2}-1260 a^{4} b^{4} p \,x^{2}-7 a^{2} b^{6} p^{6} x^{3}-105 a^{2} b^{6} p^{5} x^{3}-595 a^{2} b^{6} p^{4} x^{3}-1575 a^{2} b^{6} p^{3} x^{3}-1918 a^{2} b^{6} p^{2} x^{3}-840 a^{2} b^{6} p \,x^{3}+840 \sqrt {x}\, a^{5} b^{3} p^{3} x +2520 \sqrt {x}\, a^{5} b^{3} p^{2} x +1680 \sqrt {x}\, a^{5} b^{3} p x +42 \sqrt {x}\, a^{3} b^{5} p^{5} x^{2}+420 \sqrt {x}\, a^{3} b^{5} p^{4} x^{2}+1470 \sqrt {x}\, a^{3} b^{5} p^{3} x^{2}+2100 \sqrt {x}\, a^{3} b^{5} p^{2} x^{2}+1008 \sqrt {x}\, a^{3} b^{5} p \,x^{2}+21 \sqrt {x}\, a \,b^{7} p^{6} x^{3}+175 \sqrt {x}\, a \,b^{7} p^{5} x^{3}+735 \sqrt {x}\, a \,b^{7} p^{4} x^{3}+1624 \sqrt {x}\, a \,b^{7} p^{3} x^{3}+1764 \sqrt {x}\, a \,b^{7} p^{2} x^{3}+720 \sqrt {x}\, a \,b^{7} p \,x^{3}+28 b^{8} p^{6} x^{4}+322 b^{8} p^{5} x^{4}+1960 b^{8} p^{4} x^{4}+6769 b^{8} p^{3} x^{4}+13132 b^{8} p^{2} x^{4}+13068 b^{8} p \,x^{4}-5040 a^{8}+\sqrt {x}\, a \,b^{7} p^{7} x^{3}+5040 b^{8} x^{4}\right )}{b^{8} \left (p^{8}+36 p^{7}+546 p^{6}+4536 p^{5}+22449 p^{4}+67284 p^{3}+118124 p^{2}+109584 p +40320\right )} \] Input:

int((a+b*x^(1/2))^p*x^3,x)
 

Output:

(2*(sqrt(x)*b + a)**p*(5040*sqrt(x)*a**7*b*p + 840*sqrt(x)*a**5*b**3*p**3* 
x + 2520*sqrt(x)*a**5*b**3*p**2*x + 1680*sqrt(x)*a**5*b**3*p*x + 42*sqrt(x 
)*a**3*b**5*p**5*x**2 + 420*sqrt(x)*a**3*b**5*p**4*x**2 + 1470*sqrt(x)*a** 
3*b**5*p**3*x**2 + 2100*sqrt(x)*a**3*b**5*p**2*x**2 + 1008*sqrt(x)*a**3*b* 
*5*p*x**2 + sqrt(x)*a*b**7*p**7*x**3 + 21*sqrt(x)*a*b**7*p**6*x**3 + 175*s 
qrt(x)*a*b**7*p**5*x**3 + 735*sqrt(x)*a*b**7*p**4*x**3 + 1624*sqrt(x)*a*b* 
*7*p**3*x**3 + 1764*sqrt(x)*a*b**7*p**2*x**3 + 720*sqrt(x)*a*b**7*p*x**3 - 
 5040*a**8 - 2520*a**6*b**2*p**2*x - 2520*a**6*b**2*p*x - 210*a**4*b**4*p* 
*4*x**2 - 1260*a**4*b**4*p**3*x**2 - 2310*a**4*b**4*p**2*x**2 - 1260*a**4* 
b**4*p*x**2 - 7*a**2*b**6*p**6*x**3 - 105*a**2*b**6*p**5*x**3 - 595*a**2*b 
**6*p**4*x**3 - 1575*a**2*b**6*p**3*x**3 - 1918*a**2*b**6*p**2*x**3 - 840* 
a**2*b**6*p*x**3 + b**8*p**7*x**4 + 28*b**8*p**6*x**4 + 322*b**8*p**5*x**4 
 + 1960*b**8*p**4*x**4 + 6769*b**8*p**3*x**4 + 13132*b**8*p**2*x**4 + 1306 
8*b**8*p*x**4 + 5040*b**8*x**4))/(b**8*(p**8 + 36*p**7 + 546*p**6 + 4536*p 
**5 + 22449*p**4 + 67284*p**3 + 118124*p**2 + 109584*p + 40320))