3.7.69 \(\int (b+a x^3) \sqrt {x+x^4} \, dx\)

Optimal. Leaf size=53 \[ \frac {1}{12} \sqrt {x^4+x} \left (2 a x^4+a x+4 b x\right )+\frac {1}{12} (4 b-a) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 86, normalized size of antiderivative = 1.62, number of steps used = 9, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2053, 2004, 2029, 206, 2021, 2024} \begin {gather*} \frac {1}{6} a \sqrt {x^4+x} x^4+\frac {1}{12} a \sqrt {x^4+x} x-\frac {1}{12} a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )+\frac {1}{3} b \sqrt {x^4+x} x+\frac {1}{3} b \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + a*x^3)*Sqrt[x + x^4],x]

[Out]

(a*x*Sqrt[x + x^4])/12 + (b*x*Sqrt[x + x^4])/3 + (a*x^4*Sqrt[x + x^4])/6 - (a*ArcTanh[x^2/Sqrt[x + x^4]])/12 +
(b*ArcTanh[x^2/Sqrt[x + x^4]])/3

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2004

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(n*p + 1), x] + Dist[(
a*(n - j)*p)/(n*p + 1), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0,
j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 2053

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \left (b+a x^3\right ) \sqrt {x+x^4} \, dx &=\int \left (b \sqrt {x+x^4}+a x^3 \sqrt {x+x^4}\right ) \, dx\\ &=a \int x^3 \sqrt {x+x^4} \, dx+b \int \sqrt {x+x^4} \, dx\\ &=\frac {1}{3} b x \sqrt {x+x^4}+\frac {1}{6} a x^4 \sqrt {x+x^4}+\frac {1}{4} a \int \frac {x^4}{\sqrt {x+x^4}} \, dx+\frac {1}{2} b \int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{12} a x \sqrt {x+x^4}+\frac {1}{3} b x \sqrt {x+x^4}+\frac {1}{6} a x^4 \sqrt {x+x^4}-\frac {1}{8} a \int \frac {x}{\sqrt {x+x^4}} \, dx+\frac {1}{3} b \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=\frac {1}{12} a x \sqrt {x+x^4}+\frac {1}{3} b x \sqrt {x+x^4}+\frac {1}{6} a x^4 \sqrt {x+x^4}+\frac {1}{3} b \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )-\frac {1}{12} a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=\frac {1}{12} a x \sqrt {x+x^4}+\frac {1}{3} b x \sqrt {x+x^4}+\frac {1}{6} a x^4 \sqrt {x+x^4}-\frac {1}{12} a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )+\frac {1}{3} b \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 58, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x^4+x} \left (x^{3/2} \left (2 a x^3+a+4 b\right )-\frac {(a-4 b) \sinh ^{-1}\left (x^{3/2}\right )}{\sqrt {x^3+1}}\right )}{12 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + a*x^3)*Sqrt[x + x^4],x]

[Out]

(Sqrt[x + x^4]*(x^(3/2)*(a + 4*b + 2*a*x^3) - ((a - 4*b)*ArcSinh[x^(3/2)])/Sqrt[1 + x^3]))/(12*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.47, size = 53, normalized size = 1.00 \begin {gather*} \frac {1}{12} \sqrt {x+x^4} \left (a x+4 b x+2 a x^4\right )+\frac {1}{12} (-a+4 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + a*x^3)*Sqrt[x + x^4],x]

[Out]

(Sqrt[x + x^4]*(a*x + 4*b*x + 2*a*x^4))/12 + ((-a + 4*b)*ArcTanh[x^2/Sqrt[x + x^4]])/12

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fricas [A]  time = 0.48, size = 49, normalized size = 0.92 \begin {gather*} -\frac {1}{24} \, {\left (a - 4 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + \frac {1}{12} \, {\left (2 \, a x^{4} + {\left (a + 4 \, b\right )} x\right )} \sqrt {x^{4} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b)*(x^4+x)^(1/2),x, algorithm="fricas")

[Out]

-1/24*(a - 4*b)*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1) + 1/12*(2*a*x^4 + (a + 4*b)*x)*sqrt(x^4 + x)

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giac [A]  time = 0.28, size = 53, normalized size = 1.00 \begin {gather*} \frac {1}{12} \, {\left (2 \, a x^{3} + a + 4 \, b\right )} \sqrt {x^{4} + x} x - \frac {1}{24} \, {\left (a - 4 \, b\right )} {\left (\log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b)*(x^4+x)^(1/2),x, algorithm="giac")

[Out]

1/12*(2*a*x^3 + a + 4*b)*sqrt(x^4 + x)*x - 1/24*(a - 4*b)*(log(sqrt(1/x^3 + 1) + 1) - log(abs(sqrt(1/x^3 + 1)
- 1)))

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maple [A]  time = 0.28, size = 48, normalized size = 0.91

method	result	size
trager	\(\frac {x \left (2 a \,x^{3}+a +4 b \right ) \sqrt {x^{4}+x}}{12}-\frac {\left (a -4 b \right ) \ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{24}\)	\(48\)
meijerg	\(-\frac {a \left (-\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \left (6 x^{3}+3\right ) \sqrt {x^{3}+1}}{6}+\frac {\sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )}{2}\right )}{6 \sqrt {\pi }}-\frac {b \left (-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}\)	\(71\)
elliptic	\(\frac {a \,x^{4} \sqrt {x^{4}+x}}{6}+\left (\frac {a}{12}+\frac {b}{3}\right ) x \sqrt {x^{4}+x}-\frac {2 \left (\frac {b}{2}-\frac {a}{8}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(327\)
risch	\(\frac {x^{2} \left (2 a \,x^{3}+a +4 b \right ) \left (x^{3}+1\right )}{12 \sqrt {x \left (x^{3}+1\right )}}-\frac {2 \left (\frac {b}{2}-\frac {a}{8}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(328\)
default	\(a \left (\frac {x^{4} \sqrt {x^{4}+x}}{6}+\frac {x \sqrt {x^{4}+x}}{12}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+b \left (\frac {x \sqrt {x^{4}+x}}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\)	\(618\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3+b)*(x^4+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*x*(2*a*x^3+a+4*b)*(x^4+x)^(1/2)-1/24*(a-4*b)*ln(-2*x^3-2*x*(x^4+x)^(1/2)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (a x^{3} + b\right )} \sqrt {x^{4} + x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b)*(x^4+x)^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x^3 + b)*sqrt(x^4 + x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \left (a\,x^3+b\right )\,\sqrt {x^4+x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + a*x^3)*(x + x^4)^(1/2),x)

[Out]

int((b + a*x^3)*(x + x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{3} + b\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3+b)*(x**4+x)**(1/2),x)

[Out]

Integral(sqrt(x*(x + 1)*(x**2 - x + 1))*(a*x**3 + b), x)

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