3.526 \(\int \frac{x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=200 \[ \frac{35 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-9 a B)}{64 b^5}-\frac{35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{11/2}}-\frac{x^{7/2} \sqrt{a+b x} (8 A b-9 a B)}{4 a b^2}+\frac{7 x^{5/2} \sqrt{a+b x} (8 A b-9 a B)}{24 b^3}-\frac{35 a x^{3/2} \sqrt{a+b x} (8 A b-9 a B)}{96 b^4}+\frac{2 x^{9/2} (A b-a B)}{a b \sqrt{a+b x}} \]

[Out]

(2*(A*b - a*B)*x^(9/2))/(a*b*Sqrt[a + b*x]) + (35*a^2*(8*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^5) - (35*a*
(8*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(96*b^4) + (7*(8*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^3) - ((8*A*b
- 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(4*a*b^2) - (35*a^3*(8*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])
/(64*b^(11/2))

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Rubi [A]  time = 0.0883846, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 50, 63, 217, 206} \[ \frac{35 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-9 a B)}{64 b^5}-\frac{35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{11/2}}-\frac{x^{7/2} \sqrt{a+b x} (8 A b-9 a B)}{4 a b^2}+\frac{7 x^{5/2} \sqrt{a+b x} (8 A b-9 a B)}{24 b^3}-\frac{35 a x^{3/2} \sqrt{a+b x} (8 A b-9 a B)}{96 b^4}+\frac{2 x^{9/2} (A b-a B)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2*(A*b - a*B)*x^(9/2))/(a*b*Sqrt[a + b*x]) + (35*a^2*(8*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^5) - (35*a*
(8*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(96*b^4) + (7*(8*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^3) - ((8*A*b
- 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(4*a*b^2) - (35*a^3*(8*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])
/(64*b^(11/2))

Rule 78

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

Rule 50

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 63

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 217

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]

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])

Rubi steps

\begin{align*} \int \frac{x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}-\frac{\left (2 \left (4 A b-\frac{9 a B}{2}\right )\right ) \int \frac{x^{7/2}}{\sqrt{a+b x}} \, dx}{a b}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}+\frac{(7 (8 A b-9 a B)) \int \frac{x^{5/2}}{\sqrt{a+b x}} \, dx}{8 b^2}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}+\frac{7 (8 A b-9 a B) x^{5/2} \sqrt{a+b x}}{24 b^3}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}-\frac{(35 a (8 A b-9 a B)) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{48 b^3}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}-\frac{35 a (8 A b-9 a B) x^{3/2} \sqrt{a+b x}}{96 b^4}+\frac{7 (8 A b-9 a B) x^{5/2} \sqrt{a+b x}}{24 b^3}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}+\frac{\left (35 a^2 (8 A b-9 a B)\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{64 b^4}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}+\frac{35 a^2 (8 A b-9 a B) \sqrt{x} \sqrt{a+b x}}{64 b^5}-\frac{35 a (8 A b-9 a B) x^{3/2} \sqrt{a+b x}}{96 b^4}+\frac{7 (8 A b-9 a B) x^{5/2} \sqrt{a+b x}}{24 b^3}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}-\frac{\left (35 a^3 (8 A b-9 a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{128 b^5}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}+\frac{35 a^2 (8 A b-9 a B) \sqrt{x} \sqrt{a+b x}}{64 b^5}-\frac{35 a (8 A b-9 a B) x^{3/2} \sqrt{a+b x}}{96 b^4}+\frac{7 (8 A b-9 a B) x^{5/2} \sqrt{a+b x}}{24 b^3}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}-\frac{\left (35 a^3 (8 A b-9 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{64 b^5}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}+\frac{35 a^2 (8 A b-9 a B) \sqrt{x} \sqrt{a+b x}}{64 b^5}-\frac{35 a (8 A b-9 a B) x^{3/2} \sqrt{a+b x}}{96 b^4}+\frac{7 (8 A b-9 a B) x^{5/2} \sqrt{a+b x}}{24 b^3}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}-\frac{\left (35 a^3 (8 A b-9 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^5}\\ &=\frac{2 (A b-a B) x^{9/2}}{a b \sqrt{a+b x}}+\frac{35 a^2 (8 A b-9 a B) \sqrt{x} \sqrt{a+b x}}{64 b^5}-\frac{35 a (8 A b-9 a B) x^{3/2} \sqrt{a+b x}}{96 b^4}+\frac{7 (8 A b-9 a B) x^{5/2} \sqrt{a+b x}}{24 b^3}-\frac{(8 A b-9 a B) x^{7/2} \sqrt{a+b x}}{4 a b^2}-\frac{35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.18599, size = 151, normalized size = 0.76 \[ \frac{\frac{(a+b x) (9 a B-8 A b) \left (b x \sqrt{\frac{b x}{a}+1} \left (70 a^2 b x-105 a^3-56 a b^2 x^2+48 b^3 x^3\right )+105 a^{7/2} \sqrt{b} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{3 \sqrt{\frac{b x}{a}+1}}+128 b^5 x^5 (A b-a B)}{64 a b^6 \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(128*b^5*(A*b - a*B)*x^5 + ((-8*A*b + 9*a*B)*(a + b*x)*(b*x*Sqrt[1 + (b*x)/a]*(-105*a^3 + 70*a^2*b*x - 56*a*b^
2*x^2 + 48*b^3*x^3) + 105*a^(7/2)*Sqrt[b]*Sqrt[x]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(3*Sqrt[1 + (b*x)/a]))/
(64*a*b^6*Sqrt[x]*Sqrt[a + b*x])

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Maple [B]  time = 0.017, size = 330, normalized size = 1.7 \begin{align*} -{\frac{1}{384} \left ( -96\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-128\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+144\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+224\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-252\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+840\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}{b}^{2}-560\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x{a}^{2}-945\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{4}b+630\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{3}+840\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{4}b-1680\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{a}^{3}-945\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{5}+1890\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{4} \right ) \sqrt{x}{b}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x)

[Out]

-1/384*(-96*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)-128*A*x^3*b^(9/2)*(x*(b*x+a))^(1/2)+144*B*x^3*a*b^(7/2)*(x*(b*x+a)
)^(1/2)+224*A*x^2*a*b^(7/2)*(x*(b*x+a))^(1/2)-252*B*x^2*a^2*b^(5/2)*(x*(b*x+a))^(1/2)+840*A*ln(1/2*(2*(x*(b*x+
a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a^3*b^2-560*A*(x*(b*x+a))^(1/2)*b^(5/2)*x*a^2-945*B*ln(1/2*(2*(x*(b*x+a)
)^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a^4*b+630*B*(x*(b*x+a))^(1/2)*b^(3/2)*x*a^3+840*A*ln(1/2*(2*(x*(b*x+a))^(1
/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^4*b-1680*A*(x*(b*x+a))^(1/2)*b^(3/2)*a^3-945*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(
1/2)+2*b*x+a)/b^(1/2))*a^5+1890*B*(x*(b*x+a))^(1/2)*b^(1/2)*a^4)/b^(11/2)*x^(1/2)/(x*(b*x+a))^(1/2)/(b*x+a)^(1
/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.69339, size = 833, normalized size = 4.16 \begin{align*} \left [-\frac{105 \,{\left (9 \, B a^{5} - 8 \, A a^{4} b +{\left (9 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b} \log \left (2 \, b x - 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (48 \, B b^{5} x^{4} - 945 \, B a^{4} b + 840 \, A a^{3} b^{2} - 8 \,{\left (9 \, B a b^{4} - 8 \, A b^{5}\right )} x^{3} + 14 \,{\left (9 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{2} - 35 \,{\left (9 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{384 \,{\left (b^{7} x + a b^{6}\right )}}, -\frac{105 \,{\left (9 \, B a^{5} - 8 \, A a^{4} b +{\left (9 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (48 \, B b^{5} x^{4} - 945 \, B a^{4} b + 840 \, A a^{3} b^{2} - 8 \,{\left (9 \, B a b^{4} - 8 \, A b^{5}\right )} x^{3} + 14 \,{\left (9 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{2} - 35 \,{\left (9 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{192 \,{\left (b^{7} x + a b^{6}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/384*(105*(9*B*a^5 - 8*A*a^4*b + (9*B*a^4*b - 8*A*a^3*b^2)*x)*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*s
qrt(x) + a) - 2*(48*B*b^5*x^4 - 945*B*a^4*b + 840*A*a^3*b^2 - 8*(9*B*a*b^4 - 8*A*b^5)*x^3 + 14*(9*B*a^2*b^3 -
8*A*a*b^4)*x^2 - 35*(9*B*a^3*b^2 - 8*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^7*x + a*b^6), -1/192*(105*(9*B*a^
5 - 8*A*a^4*b + (9*B*a^4*b - 8*A*a^3*b^2)*x)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (48*B*b^5*x
^4 - 945*B*a^4*b + 840*A*a^3*b^2 - 8*(9*B*a*b^4 - 8*A*b^5)*x^3 + 14*(9*B*a^2*b^3 - 8*A*a*b^4)*x^2 - 35*(9*B*a^
3*b^2 - 8*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^7*x + a*b^6)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 88.0709, size = 340, normalized size = 1.7 \begin{align*} \frac{1}{192} \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{7}} - \frac{33 \, B a b^{27}{\left | b \right |} - 8 \, A b^{28}{\left | b \right |}}{b^{34}}\right )} + \frac{315 \, B a^{2} b^{27}{\left | b \right |} - 152 \, A a b^{28}{\left | b \right |}}{b^{34}}\right )} - \frac{3 \,{\left (325 \, B a^{3} b^{27}{\left | b \right |} - 232 \, A a^{2} b^{28}{\left | b \right |}\right )}}{b^{34}}\right )} \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a} - \frac{35 \,{\left (9 \, B a^{4} \sqrt{b}{\left | b \right |} - 8 \, A a^{3} b^{\frac{3}{2}}{\left | b \right |}\right )} \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{128 \, b^{7}} - \frac{4 \,{\left (B a^{5} \sqrt{b}{\left | b \right |} - A a^{4} b^{\frac{3}{2}}{\left | b \right |}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/192*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)*B*abs(b)/b^7 - (33*B*a*b^27*abs(b) - 8*A*b^28*abs(b))/b^34) + (31
5*B*a^2*b^27*abs(b) - 152*A*a*b^28*abs(b))/b^34) - 3*(325*B*a^3*b^27*abs(b) - 232*A*a^2*b^28*abs(b))/b^34)*sqr
t((b*x + a)*b - a*b)*sqrt(b*x + a) - 35/128*(9*B*a^4*sqrt(b)*abs(b) - 8*A*a^3*b^(3/2)*abs(b))*log((sqrt(b*x +
a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^7 - 4*(B*a^5*sqrt(b)*abs(b) - A*a^4*b^(3/2)*abs(b))/(((sqrt(b*x + a
)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*b^6)