3.3.25 \(\int \frac {x^{7/2} (A+B x)}{\sqrt {b x+c x^2}} \, dx\)

Optimal. Leaf size=170 \[ \frac {32 b^3 \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt {x}}-\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac {4 b x^{3/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{105 c^3}-\frac {2 x^{5/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c} \]

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Rubi [A]  time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {794, 656, 648} \begin {gather*} -\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac {32 b^3 \sqrt {b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt {x}}-\frac {2 x^{5/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac {4 b x^{3/2} \sqrt {b x+c x^2} (8 b B-9 A c)}{105 c^3}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32*b^3*(8*b*B - 9*A*c)*Sqrt[b*x + c*x^2])/(315*c^5*Sqrt[x]) - (16*b^2*(8*b*B - 9*A*c)*Sqrt[x]*Sqrt[b*x + c*x^
2])/(315*c^4) + (4*b*(8*b*B - 9*A*c)*x^(3/2)*Sqrt[b*x + c*x^2])/(105*c^3) - (2*(8*b*B - 9*A*c)*x^(5/2)*Sqrt[b*
x + c*x^2])/(63*c^2) + (2*B*x^(7/2)*Sqrt[b*x + c*x^2])/(9*c)

Rule 648

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

Rule 656

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

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rubi steps

\begin {align*} \int \frac {x^{7/2} (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}+\frac {\left (2 \left (\frac {7}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx}{9 c}\\ &=-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}+\frac {(2 b (8 b B-9 A c)) \int \frac {x^{5/2}}{\sqrt {b x+c x^2}} \, dx}{21 c^2}\\ &=\frac {4 b (8 b B-9 A c) x^{3/2} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}-\frac {\left (8 b^2 (8 b B-9 A c)\right ) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{105 c^3}\\ &=-\frac {16 b^2 (8 b B-9 A c) \sqrt {x} \sqrt {b x+c x^2}}{315 c^4}+\frac {4 b (8 b B-9 A c) x^{3/2} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}+\frac {\left (16 b^3 (8 b B-9 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{315 c^4}\\ &=\frac {32 b^3 (8 b B-9 A c) \sqrt {b x+c x^2}}{315 c^5 \sqrt {x}}-\frac {16 b^2 (8 b B-9 A c) \sqrt {x} \sqrt {b x+c x^2}}{315 c^4}+\frac {4 b (8 b B-9 A c) x^{3/2} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (8 b B-9 A c) x^{5/2} \sqrt {b x+c x^2}}{63 c^2}+\frac {2 B x^{7/2} \sqrt {b x+c x^2}}{9 c}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 94, normalized size = 0.55 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (-16 b^3 c (9 A+4 B x)+24 b^2 c^2 x (3 A+2 B x)-2 b c^3 x^2 (27 A+20 B x)+5 c^4 x^3 (9 A+7 B x)+128 b^4 B\right )}{315 c^5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x*(b + c*x)]*(128*b^4*B + 24*b^2*c^2*x*(3*A + 2*B*x) - 16*b^3*c*(9*A + 4*B*x) + 5*c^4*x^3*(9*A + 7*B*x
) - 2*b*c^3*x^2*(27*A + 20*B*x)))/(315*c^5*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.14, size = 107, normalized size = 0.63 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-144 A b^3 c+72 A b^2 c^2 x-54 A b c^3 x^2+45 A c^4 x^3+128 b^4 B-64 b^3 B c x+48 b^2 B c^2 x^2-40 b B c^3 x^3+35 B c^4 x^4\right )}{315 c^5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x + c*x^2]*(128*b^4*B - 144*A*b^3*c - 64*b^3*B*c*x + 72*A*b^2*c^2*x + 48*b^2*B*c^2*x^2 - 54*A*b*c^3*
x^2 - 40*b*B*c^3*x^3 + 45*A*c^4*x^3 + 35*B*c^4*x^4))/(315*c^5*Sqrt[x])

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fricas [A]  time = 0.41, size = 103, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (35 \, B c^{4} x^{4} + 128 \, B b^{4} - 144 \, A b^{3} c - 5 \, {\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{2} - 8 \, {\left (8 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, c^{5} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c^4*x^4 + 128*B*b^4 - 144*A*b^3*c - 5*(8*B*b*c^3 - 9*A*c^4)*x^3 + 6*(8*B*b^2*c^2 - 9*A*b*c^3)*x^2
- 8*(8*B*b^3*c - 9*A*b^2*c^2)*x)*sqrt(c*x^2 + b*x)/(c^5*sqrt(x))

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giac [A]  time = 0.19, size = 135, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (B b^{4} - A b^{3} c\right )} \sqrt {c x + b}}{c^{5}} + \frac {2 \, {\left (35 \, {\left (c x + b\right )}^{\frac {9}{2}} B - 180 \, {\left (c x + b\right )}^{\frac {7}{2}} B b + 378 \, {\left (c x + b\right )}^{\frac {5}{2}} B b^{2} - 420 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{3} + 45 \, {\left (c x + b\right )}^{\frac {7}{2}} A c - 189 \, {\left (c x + b\right )}^{\frac {5}{2}} A b c + 315 \, {\left (c x + b\right )}^{\frac {3}{2}} A b^{2} c\right )}}{315 \, c^{5}} - \frac {32 \, {\left (8 \, B b^{\frac {9}{2}} - 9 \, A b^{\frac {7}{2}} c\right )}}{315 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(B*b^4 - A*b^3*c)*sqrt(c*x + b)/c^5 + 2/315*(35*(c*x + b)^(9/2)*B - 180*(c*x + b)^(7/2)*B*b + 378*(c*x + b)^
(5/2)*B*b^2 - 420*(c*x + b)^(3/2)*B*b^3 + 45*(c*x + b)^(7/2)*A*c - 189*(c*x + b)^(5/2)*A*b*c + 315*(c*x + b)^(
3/2)*A*b^2*c)/c^5 - 32/315*(8*B*b^(9/2) - 9*A*b^(7/2)*c)/c^5

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maple [A]  time = 0.05, size = 107, normalized size = 0.63 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-35 B \,x^{4} c^{4}-45 A \,c^{4} x^{3}+40 B b \,c^{3} x^{3}+54 A b \,c^{3} x^{2}-48 B \,b^{2} c^{2} x^{2}-72 A \,b^{2} c^{2} x +64 B \,b^{3} c x +144 A \,b^{3} c -128 b^{4} B \right ) \sqrt {x}}{315 \sqrt {c \,x^{2}+b x}\, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(c*x+b)*(-35*B*c^4*x^4-45*A*c^4*x^3+40*B*b*c^3*x^3+54*A*b*c^3*x^2-48*B*b^2*c^2*x^2-72*A*b^2*c^2*x+64*B*
b^3*c*x+144*A*b^3*c-128*B*b^4)*x^(1/2)/c^5/(c*x^2+b*x)^(1/2)

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maxima [A]  time = 0.70, size = 120, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )} A}{35 \, \sqrt {c x + b} c^{4}} + \frac {2 \, {\left (35 \, c^{5} x^{5} - 5 \, b c^{4} x^{4} + 8 \, b^{2} c^{3} x^{3} - 16 \, b^{3} c^{2} x^{2} + 64 \, b^{4} c x + 128 \, b^{5}\right )} B}{315 \, \sqrt {c x + b} c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*c^4*x^4 - b*c^3*x^3 + 2*b^2*c^2*x^2 - 8*b^3*c*x - 16*b^4)*A/(sqrt(c*x + b)*c^4) + 2/315*(35*c^5*x^5 -
5*b*c^4*x^4 + 8*b^2*c^3*x^3 - 16*b^3*c^2*x^2 + 64*b^4*c*x + 128*b^5)*B/(sqrt(c*x + b)*c^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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