### 3.101 $$\int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=117 $-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}$

[Out]

-Sqrt[b*x + c*x^2]/(3*b*x^(7/2)) + (5*c*Sqrt[b*x + c*x^2])/(12*b^2*x^(5/2)) - (5*c^2*Sqrt[b*x + c*x^2])/(8*b^3
*x^(3/2)) + (5*c^3*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(7/2))

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Rubi [A]  time = 0.0492188, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.158, Rules used = {672, 660, 207} $-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

-Sqrt[b*x + c*x^2]/(3*b*x^(7/2)) + (5*c*Sqrt[b*x + c*x^2])/(12*b^2*x^(5/2)) - (5*c^2*Sqrt[b*x + c*x^2])/(8*b^3
*x^(3/2)) + (5*c^3*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(7/2))

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), I
nt[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}-\frac{(5 c) \int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx}{6 b}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}+\frac{\left (5 c^2\right ) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}-\frac{\left (5 c^3\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{16 b^3}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}-\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{8 b^3}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0121609, size = 40, normalized size = 0.34 $\frac{2 c^3 \sqrt{x (b+c x)} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{c x}{b}+1\right )}{b^4 \sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(2*c^3*Sqrt[x*(b + c*x)]*Hypergeometric2F1[1/2, 4, 3/2, 1 + (c*x)/b])/(b^4*Sqrt[x])

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Maple [A]  time = 0.189, size = 90, normalized size = 0.8 \begin{align*}{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-15\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+10\,x{b}^{3/2}c\sqrt{cx+b}-8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(c*x^2+b*x)^(1/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)/b^(7/2)*(15*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3-15*x^2*c^2*b^(1/2)*(c*x+b)^(1/2)+10*
x*b^(3/2)*c*(c*x+b)^(1/2)-8*b^(5/2)*(c*x+b)^(1/2))/x^(7/2)/(c*x+b)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x} x^{\frac{7}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*x^(7/2)), x)

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Fricas [A]  time = 2.18035, size = 428, normalized size = 3.66 \begin{align*} \left [\frac{15 \, \sqrt{b} c^{3} x^{4} \log \left (-\frac{c x^{2} + 2 \, b x + 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) - 2 \,{\left (15 \, b c^{2} x^{2} - 10 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{48 \, b^{4} x^{4}}, -\frac{15 \, \sqrt{-b} c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (15 \, b c^{2} x^{2} - 10 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \, b^{4} x^{4}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*sqrt(b)*c^3*x^4*log(-(c*x^2 + 2*b*x + 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) - 2*(15*b*c^2*x^2 -
10*b^2*c*x + 8*b^3)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^4*x^4), -1/24*(15*sqrt(-b)*c^3*x^4*arctan(sqrt(-b)*sqrt(x)/s
qrt(c*x^2 + b*x)) + (15*b*c^2*x^2 - 10*b^2*c*x + 8*b^3)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^4*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{7}{2}} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(x**(7/2)*sqrt(x*(b + c*x))), x)

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Giac [A]  time = 1.32539, size = 97, normalized size = 0.83 \begin{align*} -\frac{1}{24} \, c^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x + b\right )}^{\frac{5}{2}} - 40 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 33 \, \sqrt{c x + b} b^{2}}{b^{3} c^{3} x^{3}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*c^3*(15*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(c*x + b)^(5/2) - 40*(c*x + b)^(3/2)*b + 33*
sqrt(c*x + b)*b^2)/(b^3*c^3*x^3))