3.2264 \(\int (a+b \sqrt {x})^p x^3 \, dx\)

Optimal. Leaf size=204 \[ -\frac {2 a^7 \left (a+b \sqrt {x}\right )^{p+1}}{b^8 (p+1)}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{p+2}}{b^8 (p+2)}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{p+3}}{b^8 (p+3)}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{p+4}}{b^8 (p+4)}-\frac {70 a^3 \left (a+b \sqrt {x}\right )^{p+5}}{b^8 (p+5)}+\frac {42 a^2 \left (a+b \sqrt {x}\right )^{p+6}}{b^8 (p+6)}-\frac {14 a \left (a+b \sqrt {x}\right )^{p+7}}{b^8 (p+7)}+\frac {2 \left (a+b \sqrt {x}\right )^{p+8}}{b^8 (p+8)} \]

[Out]

-2*a^7*(a+b*x^(1/2))^(1+p)/b^8/(1+p)+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)

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Rubi [A]  time = 0.12, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a^7 \left (a+b \sqrt {x}\right )^{p+1}}{b^8 (p+1)}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{p+2}}{b^8 (p+2)}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{p+3}}{b^8 (p+3)}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{p+4}}{b^8 (p+4)}-\frac {70 a^3 \left (a+b \sqrt {x}\right )^{p+5}}{b^8 (p+5)}+\frac {42 a^2 \left (a+b \sqrt {x}\right )^{p+6}}{b^8 (p+6)}-\frac {14 a \left (a+b \sqrt {x}\right )^{p+7}}{b^8 (p+7)}+\frac {2 \left (a+b \sqrt {x}\right )^{p+8}}{b^8 (p+8)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^p*x^3,x]

[Out]

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

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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps

\begin {align*} \int \left (a+b \sqrt {x}\right )^p x^3 \, dx &=2 \operatorname {Subst}\left (\int x^7 (a+b x)^p \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^7 (a+b x)^p}{b^7}+\frac {7 a^6 (a+b x)^{1+p}}{b^7}-\frac {21 a^5 (a+b x)^{2+p}}{b^7}+\frac {35 a^4 (a+b x)^{3+p}}{b^7}-\frac {35 a^3 (a+b x)^{4+p}}{b^7}+\frac {21 a^2 (a+b x)^{5+p}}{b^7}-\frac {7 a (a+b x)^{6+p}}{b^7}+\frac {(a+b x)^{7+p}}{b^7}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\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)}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 168, normalized size = 0.82 \[ \frac {2 \left (-\frac {a^7}{p+1}+\frac {7 a^6 \left (a+b \sqrt {x}\right )}{p+2}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^2}{p+3}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^3}{p+4}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^4}{p+5}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^5}{p+6}-\frac {7 a \left (a+b \sqrt {x}\right )^6}{p+7}+\frac {\left (a+b \sqrt {x}\right )^7}{p+8}\right ) \left (a+b \sqrt {x}\right )^{p+1}}{b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^p*x^3,x]

[Out]

(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*Sqr
t[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

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fricas [B]  time = 1.04, size = 458, normalized size = 2.25 \[ -\frac {2 \, {\left (5040 \, a^{8} - {\left (b^{8} p^{7} + 28 \, b^{8} p^{6} + 322 \, b^{8} p^{5} + 1960 \, b^{8} p^{4} + 6769 \, b^{8} p^{3} + 13132 \, b^{8} p^{2} + 13068 \, b^{8} p + 5040 \, b^{8}\right )} x^{4} + 7 \, {\left (a^{2} b^{6} p^{6} + 15 \, a^{2} b^{6} p^{5} + 85 \, a^{2} b^{6} p^{4} + 225 \, a^{2} b^{6} p^{3} + 274 \, a^{2} b^{6} p^{2} + 120 \, a^{2} b^{6} p\right )} x^{3} + 210 \, {\left (a^{4} b^{4} p^{4} + 6 \, a^{4} b^{4} p^{3} + 11 \, a^{4} b^{4} p^{2} + 6 \, a^{4} b^{4} p\right )} x^{2} + 2520 \, {\left (a^{6} b^{2} p^{2} + a^{6} b^{2} p\right )} x - {\left (5040 \, a^{7} b p + {\left (a b^{7} p^{7} + 21 \, a b^{7} p^{6} + 175 \, a b^{7} p^{5} + 735 \, a b^{7} p^{4} + 1624 \, a b^{7} p^{3} + 1764 \, a b^{7} p^{2} + 720 \, a b^{7} p\right )} x^{3} + 42 \, {\left (a^{3} b^{5} p^{5} + 10 \, a^{3} b^{5} p^{4} + 35 \, a^{3} b^{5} p^{3} + 50 \, a^{3} b^{5} p^{2} + 24 \, a^{3} b^{5} p\right )} x^{2} + 840 \, {\left (a^{5} b^{3} p^{3} + 3 \, a^{5} b^{3} p^{2} + 2 \, a^{5} b^{3} p\right )} x\right )} \sqrt {x}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{b^{8} p^{8} + 36 \, b^{8} p^{7} + 546 \, b^{8} p^{6} + 4536 \, b^{8} p^{5} + 22449 \, b^{8} p^{4} + 67284 \, b^{8} p^{3} + 118124 \, b^{8} p^{2} + 109584 \, b^{8} p + 40320 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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giac [B]  time = 0.25, size = 1642, normalized size = 8.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*sqrt(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*sq
rt(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*sq
rt(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*sqrt(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 + 25060*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^6*p^4 - 4025*(b*sqr
t(x) + a)*(b*sqrt(x) + a)^p*a^7*p^4 + 6769*(b*sqrt(x) + a)^8*(b*sqrt(x) + a)^p*p^3 - 52528*(b*sqrt(x) + a)^7*(
b*sqrt(x) + a)^p*a*p^3 + 176589*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*a^2*p^3 - 334040*(b*sqrt(x) + a)^5*(b*sqrt
(x) + a)^p*a^3*p^3 + 384755*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^4*p^3 - 270144*(b*sqrt(x) + a)^3*(b*sqrt(x)
+ a)^p*a^5*p^3 + 107023*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^6*p^3 - 18424*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*
a^7*p^3 + 13132*(b*sqrt(x) + a)^8*(b*sqrt(x) + a)^p*p^2 - 103292*(b*sqrt(x) + a)^7*(b*sqrt(x) + a)^p*a*p^2 + 3
53430*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*a^2*p^2 - 684740*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a^3*p^2 + 81592
0*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^4*p^2 - 602532*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^5*p^2 + 256942*(b
*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^6*p^2 - 48860*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^7*p^2 + 13068*(b*sqrt(x)
+ a)^8*(b*sqrt(x) + a)^p*p - 103824*(b*sqrt(x) + a)^7*(b*sqrt(x) + a)^p*a*p + 360024*(b*sqrt(x) + a)^6*(b*sqr
t(x) + a)^p*a^2*p - 710640*(b*sqrt(x) + a)^5*(b*sqrt(x) + a)^p*a^3*p + 870660*(b*sqrt(x) + a)^4*(b*sqrt(x) + a
)^p*a^4*p - 673008*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p*a^5*p + 312984*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^6*
p - 69264*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^7*p + 5040*(b*sqrt(x) + a)^8*(b*sqrt(x) + a)^p - 40320*(b*sqrt(x
) + a)^7*(b*sqrt(x) + a)^p*a + 141120*(b*sqrt(x) + a)^6*(b*sqrt(x) + a)^p*a^2 - 282240*(b*sqrt(x) + a)^5*(b*sq
rt(x) + a)^p*a^3 + 352800*(b*sqrt(x) + a)^4*(b*sqrt(x) + a)^p*a^4 - 282240*(b*sqrt(x) + a)^3*(b*sqrt(x) + a)^p
*a^5 + 141120*(b*sqrt(x) + a)^2*(b*sqrt(x) + a)^p*a^6 - 40320*(b*sqrt(x) + a)*(b*sqrt(x) + a)^p*a^7)/((b^7*p^8
+ 36*b^7*p^7 + 546*b^7*p^6 + 4536*b^7*p^5 + 22449*b^7*p^4 + 67284*b^7*p^3 + 118124*b^7*p^2 + 109584*b^7*p + 4
0320*b^7)*b)

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int x^{3} \left (b \sqrt {x}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.91, size = 286, normalized size = 1.40 \[ \frac {2 \, {\left ({\left (p^{7} + 28 \, p^{6} + 322 \, p^{5} + 1960 \, p^{4} + 6769 \, p^{3} + 13132 \, p^{2} + 13068 \, p + 5040\right )} b^{8} x^{4} + {\left (p^{7} + 21 \, p^{6} + 175 \, p^{5} + 735 \, p^{4} + 1624 \, p^{3} + 1764 \, p^{2} + 720 \, p\right )} a b^{7} x^{\frac {7}{2}} - 7 \, {\left (p^{6} + 15 \, p^{5} + 85 \, p^{4} + 225 \, p^{3} + 274 \, p^{2} + 120 \, p\right )} a^{2} b^{6} x^{3} + 42 \, {\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} a^{3} b^{5} x^{\frac {5}{2}} - 210 \, {\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} a^{4} b^{4} x^{2} + 840 \, {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a^{5} b^{3} x^{\frac {3}{2}} - 2520 \, {\left (p^{2} + p\right )} a^{6} b^{2} x + 5040 \, a^{7} b p \sqrt {x} - 5040 \, a^{8}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{{\left (p^{8} + 36 \, p^{7} + 546 \, p^{6} + 4536 \, p^{5} + 22449 \, p^{4} + 67284 \, p^{3} + 118124 \, p^{2} + 109584 \, p + 40320\right )} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((p^7 + 28*p^6 + 322*p^5 + 1960*p^4 + 6769*p^3 + 13132*p^2 + 13068*p + 5040)*b^8*x^4 + (p^7 + 21*p^6 + 175*p
^5 + 735*p^4 + 1624*p^3 + 1764*p^2 + 720*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) - 50
40*a^8)*(b*sqrt(x) + a)^p/((p^8 + 36*p^7 + 546*p^6 + 4536*p^5 + 22449*p^4 + 67284*p^3 + 118124*p^2 + 109584*p
+ 40320)*b^8)

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mupad [B]  time = 2.11, size = 590, normalized size = 2.89 \[ {\left (a+b\,\sqrt {x}\right )}^p\,\left (\frac {2\,x^4\,\left (p^7+28\,p^6+322\,p^5+1960\,p^4+6769\,p^3+13132\,p^2+13068\,p+5040\right )}{p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320}-\frac {10080\,a^8}{b^8\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {10080\,a^7\,p\,\sqrt {x}}{b^7\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {5040\,a^6\,p\,x\,\left (p+1\right )}{b^6\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {14\,a^2\,p\,x^3\,\left (p^5+15\,p^4+85\,p^3+225\,p^2+274\,p+120\right )}{b^2\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {1680\,a^5\,p\,x^{3/2}\,\left (p^2+3\,p+2\right )}{b^5\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {420\,a^4\,p\,x^2\,\left (p^3+6\,p^2+11\,p+6\right )}{b^4\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {2\,a\,p\,x^{7/2}\,\left (p^6+21\,p^5+175\,p^4+735\,p^3+1624\,p^2+1764\,p+720\right )}{b\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {84\,a^3\,p\,x^{5/2}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}{b^3\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^(1/2))^p,x)

[Out]

(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*(10958
4*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 + 2
2449*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 + 546*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 + 67284*p^3 + 22449*p^4 + 4536*p^5 + 546*p^6 + 36*p^7 + p^8 + 40320
)))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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