3.6.14 \(\int \frac {x^{7/2} (A+B x)}{\sqrt {a+b x}} \, dx\) [514]

Optimal. Leaf size=192 \[ -\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}} \]

[Out]

7/128*a^4*(10*A*b-9*B*a)*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(11/2)+7/192*a^2*(10*A*b-9*B*a)*x^(3/2)*(b*x
+a)^(1/2)/b^4-7/240*a*(10*A*b-9*B*a)*x^(5/2)*(b*x+a)^(1/2)/b^3+1/40*(10*A*b-9*B*a)*x^(7/2)*(b*x+a)^(1/2)/b^2+1
/5*B*x^(9/2)*(b*x+a)^(1/2)/b-7/128*a^3*(10*A*b-9*B*a)*x^(1/2)*(b*x+a)^(1/2)/b^5

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \begin {gather*} \frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}-\frac {7 a^3 \sqrt {x} \sqrt {a+b x} (10 A b-9 a B)}{128 b^5}+\frac {7 a^2 x^{3/2} \sqrt {a+b x} (10 A b-9 a B)}{192 b^4}-\frac {7 a x^{5/2} \sqrt {a+b x} (10 A b-9 a B)}{240 b^3}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-9 a B)}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/Sqrt[a + b*x],x]

[Out]

(-7*a^3*(10*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^5) + (7*a^2*(10*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(19
2*b^4) - (7*a*(10*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(240*b^3) + ((10*A*b - 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(40
*b^2) + (B*x^(9/2)*Sqrt[a + b*x])/(5*b) + (7*a^4*(10*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(1
28*b^(11/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {x^{7/2} (A+B x)}{\sqrt {a+b x}} \, dx &=\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (5 A b-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{\sqrt {a+b x}} \, dx}{5 b}\\ &=\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}-\frac {(7 a (10 A b-9 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{80 b^2}\\ &=-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^2 (10 A b-9 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{96 b^3}\\ &=\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}-\frac {\left (7 a^3 (10 A b-9 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^4}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.26, size = 138, normalized size = 0.72 \begin {gather*} \frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (945 a^4 B-210 a^3 b (5 A+3 B x)+96 b^4 x^3 (5 A+4 B x)+28 a^2 b^2 x (25 A+18 B x)-16 a b^3 x^2 (35 A+27 B x)\right )+105 a^4 (-10 A b+9 a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{1920 b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x))/Sqrt[a + b*x],x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(945*a^4*B - 210*a^3*b*(5*A + 3*B*x) + 96*b^4*x^3*(5*A + 4*B*x) + 28*a^2*b^2*x*
(25*A + 18*B*x) - 16*a*b^3*x^2*(35*A + 27*B*x)) + 105*a^4*(-10*A*b + 9*a*B)*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a +
b*x]])/(1920*b^(11/2))

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 260, normalized size = 1.35

method	result	size
risch	\(-\frac {\left (-384 B \,x^{4} b^{4}-480 A \,b^{4} x^{3}+432 B a \,b^{3} x^{3}+560 A a \,b^{3} x^{2}-504 B \,a^{2} b^{2} x^{2}-700 A \,a^{2} b^{2} x +630 B \,a^{3} b x +1050 A \,a^{3} b -945 a^{4} B \right ) \sqrt {b x +a}\, \sqrt {x}}{1920 b^{5}}+\frac {\left (\frac {35 a^{4} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) A}{128 b^{\frac {9}{2}}}-\frac {63 a^{5} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B}{256 b^{\frac {11}{2}}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\)	\(186\)
default	\(\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (768 B \,b^{\frac {9}{2}} x^{4} \sqrt {\left (b x +a \right ) x}+960 A \,b^{\frac {9}{2}} x^{3} \sqrt {\left (b x +a \right ) x}-864 B a \,b^{\frac {7}{2}} x^{3} \sqrt {\left (b x +a \right ) x}-1120 A a \,b^{\frac {7}{2}} x^{2} \sqrt {\left (b x +a \right ) x}+1008 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {\left (b x +a \right ) x}+1400 A \,b^{\frac {5}{2}} \sqrt {\left (b x +a \right ) x}\, a^{2} x -1260 B \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a^{3} x +1050 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -2100 A \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a^{3}-945 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+1890 B \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\, a^{4}\right )}{3840 b^{\frac {11}{2}} \sqrt {\left (b x +a \right ) x}}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3840*x^(1/2)*(b*x+a)^(1/2)/b^(11/2)*(768*B*b^(9/2)*x^4*((b*x+a)*x)^(1/2)+960*A*b^(9/2)*x^3*((b*x+a)*x)^(1/2)
-864*B*a*b^(7/2)*x^3*((b*x+a)*x)^(1/2)-1120*A*a*b^(7/2)*x^2*((b*x+a)*x)^(1/2)+1008*B*a^2*b^(5/2)*x^2*((b*x+a)*
x)^(1/2)+1400*A*b^(5/2)*((b*x+a)*x)^(1/2)*a^2*x-1260*B*b^(3/2)*((b*x+a)*x)^(1/2)*a^3*x+1050*A*ln(1/2*(2*((b*x+
a)*x)^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^4*b-2100*A*b^(3/2)*((b*x+a)*x)^(1/2)*a^3-945*B*ln(1/2*(2*((b*x+a)*x)^(
1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^5+1890*B*b^(1/2)*((b*x+a)*x)^(1/2)*a^4)/((b*x+a)*x)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 252, normalized size = 1.31 \begin {gather*} \frac {\sqrt {b x^{2} + a x} B x^{4}}{5 \, b} - \frac {9 \, \sqrt {b x^{2} + a x} B a x^{3}}{40 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} A x^{3}}{4 \, b} + \frac {21 \, \sqrt {b x^{2} + a x} B a^{2} x^{2}}{80 \, b^{3}} - \frac {7 \, \sqrt {b x^{2} + a x} A a x^{2}}{24 \, b^{2}} - \frac {21 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{4}} + \frac {35 \, \sqrt {b x^{2} + a x} A a^{2} x}{96 \, b^{3}} - \frac {63 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {9}{2}}} + \frac {63 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

1/5*sqrt(b*x^2 + a*x)*B*x^4/b - 9/40*sqrt(b*x^2 + a*x)*B*a*x^3/b^2 + 1/4*sqrt(b*x^2 + a*x)*A*x^3/b + 21/80*sqr
t(b*x^2 + a*x)*B*a^2*x^2/b^3 - 7/24*sqrt(b*x^2 + a*x)*A*a*x^2/b^2 - 21/64*sqrt(b*x^2 + a*x)*B*a^3*x/b^4 + 35/9
6*sqrt(b*x^2 + a*x)*A*a^2*x/b^3 - 63/256*B*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 35/128*
A*a^4*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) + 63/128*sqrt(b*x^2 + a*x)*B*a^4/b^5 - 35/64*sqrt(b
*x^2 + a*x)*A*a^3/b^4

________________________________________________________________________________________

Fricas [A]
time = 1.16, size = 296, normalized size = 1.54 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (384 \, B b^{5} x^{4} + 945 \, B a^{4} b - 1050 \, A a^{3} b^{2} - 48 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3840 \, b^{6}}, \frac {105 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (384 \, B b^{5} x^{4} + 945 \, B a^{4} b - 1050 \, A a^{3} b^{2} - 48 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1920 \, b^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/3840*(105*(9*B*a^5 - 10*A*a^4*b)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(384*B*b^5*x
^4 + 945*B*a^4*b - 1050*A*a^3*b^2 - 48*(9*B*a*b^4 - 10*A*b^5)*x^3 + 56*(9*B*a^2*b^3 - 10*A*a*b^4)*x^2 - 70*(9*
B*a^3*b^2 - 10*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^6, 1/1920*(105*(9*B*a^5 - 10*A*a^4*b)*sqrt(-b)*arctan(sq
rt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (384*B*b^5*x^4 + 945*B*a^4*b - 1050*A*a^3*b^2 - 48*(9*B*a*b^4 - 10*A*b^5)*
x^3 + 56*(9*B*a^2*b^3 - 10*A*a*b^4)*x^2 - 70*(9*B*a^3*b^2 - 10*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^6]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)/(b*x+a)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-4,[1
,0,0]%%%}+%%%{-4,[0,1,1]%%%}+%%%{-4,[0,1,0]%%%}+%%%{-4,[0,0,1]%%%},0,%%%{6,[2,0,0]%%%}+%%%{12,[1,1,1]%%%}+%%%{
4,[1,1,0]%%%}+%%%{4,[

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{7/2}\,\left (A+B\,x\right )}{\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(A + B*x))/(a + b*x)^(1/2),x)

[Out]

int((x^(7/2)*(A + B*x))/(a + b*x)^(1/2), x)

________________________________________________________________________________________