3.2.3 \(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{x^7} \, dx\) [103]

Optimal. Leaf size=119 \[ -\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \]

[Out]

-2/3*B*c*(c*x^2+b*x)^(3/2)/x^3-2/5*B*(c*x^2+b*x)^(5/2)/x^5-2/7*A*(c*x^2+b*x)^(7/2)/b/x^7+2*B*c^(5/2)*arctanh(x
*c^(1/2)/(c*x^2+b*x)^(1/2))-2*B*c^2*(c*x^2+b*x)^(1/2)/x

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Rubi [A]
time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {806, 676, 634, 212} \begin {gather*} -\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^7,x]

[Out]

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

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 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 676

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^7} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+B \int \frac {\left (b x+c x^2\right )^{5/2}}{x^6} \, dx\\ &=-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+(B c) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (B c^2\right ) \int \frac {\sqrt {b x+c x^2}}{x^2} \, dx\\ &=-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (B c^3\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (2 B c^3\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {2 B c^2 \sqrt {b x+c x^2}}{x}-\frac {2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 113, normalized size = 0.95 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (\sqrt {b+c x} \left (15 A (b+c x)^3+7 b B x \left (3 b^2+11 b c x+23 c^2 x^2\right )\right )+105 b B c^{5/2} x^{7/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{105 b x^4 \sqrt {b+c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^7,x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(Sqrt[b + c*x]*(15*A*(b + c*x)^3 + 7*b*B*x*(3*b^2 + 11*b*c*x + 23*c^2*x^2)) + 105*b*B*c^
(5/2)*x^(7/2)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]]))/(105*b*x^4*Sqrt[b + c*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(99)=198\).
time = 0.54, size = 257, normalized size = 2.16

method	result	size
risch	\(-\frac {2 \left (c x +b \right ) \left (15 A \,c^{3} x^{3}+161 B b \,c^{2} x^{3}+45 A b \,c^{2} x^{2}+77 B \,b^{2} c \,x^{2}+45 A \,b^{2} c x +21 B \,b^{3} x +15 A \,b^{3}\right )}{105 x^{3} \sqrt {x \left (c x +b \right )}\, b}+B \,c^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )\)	\(114\)
default	\(B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{5 b \,x^{6}}+\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{3 b \,x^{5}}+\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{b \,x^{4}}+\frac {6 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{b \,x^{3}}-\frac {8 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{3 b \,x^{2}}-\frac {10 c \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5}+\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2}\right )}{3 b}\right )}{b}\right )}{b}\right )}{3 b}\right )}{5 b}\right )-\frac {2 A \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 b \,x^{7}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(5/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

B*(-2/5/b/x^6*(c*x^2+b*x)^(7/2)+2/5*c/b*(-2/3/b/x^5*(c*x^2+b*x)^(7/2)+4/3*c/b*(-2/b/x^4*(c*x^2+b*x)^(7/2)+6*c/
b*(2/b/x^3*(c*x^2+b*x)^(7/2)-8*c/b*(2/3/b/x^2*(c*x^2+b*x)^(7/2)-10/3*c/b*(1/5*(c*x^2+b*x)^(5/2)+1/2*b*(1/8*(2*
c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x)^(1/2))))))))))-2/7*A*(c*x^2+b*x)^(7/2)/b/x^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (99) = 198\).
time = 0.27, size = 258, normalized size = 2.17 \begin {gather*} B c^{\frac {5}{2}} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {38 \, \sqrt {c x^{2} + b x} B c^{2}}{15 \, x} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{7 \, b x} - \frac {7 \, \sqrt {c x^{2} + b x} B b c}{30 \, x^{2}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{7 \, x^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} B b^{2}}{10 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B c}{3 \, x^{3}} - \frac {3 \, \sqrt {c x^{2} + b x} A b c}{28 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{2 \, x^{4}} - \frac {15 \, \sqrt {c x^{2} + b x} A b^{2}}{28 \, x^{4}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{5 \, x^{5}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{4 \, x^{5}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^7,x, algorithm="maxima")

[Out]

B*c^(5/2)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 38/15*sqrt(c*x^2 + b*x)*B*c^2/x - 2/7*sqrt(c*x^2 + b*
x)*A*c^3/(b*x) - 7/30*sqrt(c*x^2 + b*x)*B*b*c/x^2 + 1/7*sqrt(c*x^2 + b*x)*A*c^2/x^2 + 3/10*sqrt(c*x^2 + b*x)*B
*b^2/x^3 - 1/3*(c*x^2 + b*x)^(3/2)*B*c/x^3 - 3/28*sqrt(c*x^2 + b*x)*A*b*c/x^3 - 1/2*(c*x^2 + b*x)^(3/2)*B*b/x^
4 - 15/28*sqrt(c*x^2 + b*x)*A*b^2/x^4 - 1/5*(c*x^2 + b*x)^(5/2)*B/x^5 + 5/4*(c*x^2 + b*x)^(3/2)*A*b/x^5 - (c*x
^2 + b*x)^(5/2)*A/x^6

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Fricas [A]
time = 3.91, size = 238, normalized size = 2.00 \begin {gather*} \left [\frac {105 \, B b c^{\frac {5}{2}} x^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (15 \, A b^{3} + {\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} + {\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, b x^{4}}, -\frac {2 \, {\left (105 \, B b \sqrt {-c} c^{2} x^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (15 \, A b^{3} + {\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} + {\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{105 \, b x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^7,x, algorithm="fricas")

[Out]

[1/105*(105*B*b*c^(5/2)*x^4*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(15*A*b^3 + (161*B*b*c^2 + 15*A*c
^3)*x^3 + (77*B*b^2*c + 45*A*b*c^2)*x^2 + 3*(7*B*b^3 + 15*A*b^2*c)*x)*sqrt(c*x^2 + b*x))/(b*x^4), -2/105*(105*
B*b*sqrt(-c)*c^2*x^4*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (15*A*b^3 + (161*B*b*c^2 + 15*A*c^3)*x^3 + (77
*B*b^2*c + 45*A*b*c^2)*x^2 + 3*(7*B*b^3 + 15*A*b^2*c)*x)*sqrt(c*x^2 + b*x))/(b*x^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**7,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**7, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (99) = 198\).
time = 0.80, size = 390, normalized size = 3.28 \begin {gather*} -B c^{\frac {5}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \frac {2 \, {\left (315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b c^{\frac {5}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A c^{\frac {7}{2}} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{2} c^{2} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b c^{3} + 245 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{3} c^{\frac {3}{2}} + 525 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{2} c^{\frac {5}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{4} c + 525 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{3} c^{2} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{5} \sqrt {c} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{4} c^{\frac {3}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{5} c + 15 \, A b^{6} \sqrt {c}\right )}}{105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^7,x, algorithm="giac")

[Out]

-B*c^(5/2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)) + 2/105*(315*(sqrt(c)*x - sqrt(c*x^2 + b*x
))^6*B*b*c^(5/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*c^(7/2) + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b
^2*c^2 + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^3 + 245*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c^(3/2) +
525*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(5/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c + 525*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^2 + 21*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*sqrt(c) + 315*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^2*A*b^4*c^(3/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*c + 15*A*b^6*sqrt(c))/((sqrt
(c)*x - sqrt(c*x^2 + b*x))^7*sqrt(c))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(5/2)*(A + B*x))/x^7,x)

[Out]

int(((b*x + c*x^2)^(5/2)*(A + B*x))/x^7, x)

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