3.2243 \(\int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{16 b^2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e \sqrt{d+e x} (b d-a e)^4}+\frac{8 b \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x
])/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*b*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^3
*(d + e*x)^(3/2)) + (16*b^2*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^4*Sqrt[d + e*x])

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Rubi [A]  time = 0.126444, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac{16 b^2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e \sqrt{d+e x} (b d-a e)^4}+\frac{8 b \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(-2*(B*d - A*e)*Sqrt[a + b*x])/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x
])/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2)) + (8*b*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^3
*(d + e*x)^(3/2)) + (16*b^2*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x])/(105*e*(b*d - a*e)^4*Sqrt[d + e*x])

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 45

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

Rule 37

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

Rubi steps

\begin{align*} \int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx &=-\frac{2 (B d-A e) \sqrt{a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{(b B d+6 A b e-7 a B e) \int \frac{1}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{7 e (b d-a e)}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{2 (b B d+6 A b e-7 a B e) \sqrt{a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac{(4 b (b B d+6 A b e-7 a B e)) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{35 e (b d-a e)^2}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{2 (b B d+6 A b e-7 a B e) \sqrt{a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac{8 b (b B d+6 A b e-7 a B e) \sqrt{a+b x}}{105 e (b d-a e)^3 (d+e x)^{3/2}}+\frac{\left (8 b^2 (b B d+6 A b e-7 a B e)\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{105 e (b d-a e)^3}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{2 (b B d+6 A b e-7 a B e) \sqrt{a+b x}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac{8 b (b B d+6 A b e-7 a B e) \sqrt{a+b x}}{105 e (b d-a e)^3 (d+e x)^{3/2}}+\frac{16 b^2 (b B d+6 A b e-7 a B e) \sqrt{a+b x}}{105 e (b d-a e)^4 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.22981, size = 113, normalized size = 0.57 \[ \frac{2 \sqrt{a+b x} \left (15 (B d-A e)-\frac{(d+e x) \left (4 b (d+e x) (-a e+3 b d+2 b e x)+3 (b d-a e)^2\right ) (-7 a B e+6 A b e+b B d)}{(b d-a e)^3}\right )}{105 e (d+e x)^{7/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(2*Sqrt[a + b*x]*(15*(B*d - A*e) - ((b*B*d + 6*A*b*e - 7*a*B*e)*(d + e*x)*(3*(b*d - a*e)^2 + 4*b*(d + e*x)*(3*
b*d - a*e + 2*b*e*x)))/(b*d - a*e)^3))/(105*e*(-(b*d) + a*e)*(d + e*x)^(7/2))

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Maple [A]  time = 0.008, size = 322, normalized size = 1.6 \begin{align*} -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+112\,Ba{b}^{2}{e}^{3}{x}^{3}-16\,B{b}^{3}d{e}^{2}{x}^{3}+48\,Aa{b}^{2}{e}^{3}{x}^{2}-336\,A{b}^{3}d{e}^{2}{x}^{2}-56\,B{a}^{2}b{e}^{3}{x}^{2}+400\,Ba{b}^{2}d{e}^{2}{x}^{2}-56\,B{b}^{3}{d}^{2}e{x}^{2}-36\,A{a}^{2}b{e}^{3}x+168\,Aa{b}^{2}d{e}^{2}x-420\,A{b}^{3}{d}^{2}ex+42\,B{a}^{3}{e}^{3}x-202\,B{a}^{2}bd{e}^{2}x+518\,Ba{b}^{2}{d}^{2}ex-70\,B{b}^{3}{d}^{3}x+30\,A{a}^{3}{e}^{3}-126\,A{a}^{2}bd{e}^{2}+210\,Aa{b}^{2}{d}^{2}e-210\,A{b}^{3}{d}^{3}+12\,B{a}^{3}d{e}^{2}-56\,B{a}^{2}b{d}^{2}e+140\,Ba{b}^{2}{d}^{3}}{105\,{e}^{4}{a}^{4}-420\,b{e}^{3}d{a}^{3}+630\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-420\,a{b}^{3}{d}^{3}e+105\,{b}^{4}{d}^{4}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x)

[Out]

-2/105*(b*x+a)^(1/2)*(-48*A*b^3*e^3*x^3+56*B*a*b^2*e^3*x^3-8*B*b^3*d*e^2*x^3+24*A*a*b^2*e^3*x^2-168*A*b^3*d*e^
2*x^2-28*B*a^2*b*e^3*x^2+200*B*a*b^2*d*e^2*x^2-28*B*b^3*d^2*e*x^2-18*A*a^2*b*e^3*x+84*A*a*b^2*d*e^2*x-210*A*b^
3*d^2*e*x+21*B*a^3*e^3*x-101*B*a^2*b*d*e^2*x+259*B*a*b^2*d^2*e*x-35*B*b^3*d^3*x+15*A*a^3*e^3-63*A*a^2*b*d*e^2+
105*A*a*b^2*d^2*e-105*A*b^3*d^3+6*B*a^3*d*e^2-28*B*a^2*b*d^2*e+70*B*a*b^2*d^3)/(e*x+d)^(7/2)/(a^4*e^4-4*a^3*b*
d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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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((B*x+A)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 56.2094, size = 1131, normalized size = 5.71 \begin{align*} -\frac{2 \,{\left (15 \, A a^{3} e^{3} + 35 \,{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} d^{3} - 7 \,{\left (4 \, B a^{2} b - 15 \, A a b^{2}\right )} d^{2} e + 3 \,{\left (2 \, B a^{3} - 21 \, A a^{2} b\right )} d e^{2} - 8 \,{\left (B b^{3} d e^{2} -{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \,{\left (7 \, B b^{3} d^{2} e - 2 \,{\left (25 \, B a b^{2} - 21 \, A b^{3}\right )} d e^{2} +{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} -{\left (35 \, B b^{3} d^{3} - 7 \,{\left (37 \, B a b^{2} - 30 \, A b^{3}\right )} d^{2} e +{\left (101 \, B a^{2} b - 84 \, A a b^{2}\right )} d e^{2} - 3 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{105 \,{\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(15*A*a^3*e^3 + 35*(2*B*a*b^2 - 3*A*b^3)*d^3 - 7*(4*B*a^2*b - 15*A*a*b^2)*d^2*e + 3*(2*B*a^3 - 21*A*a^2
*b)*d*e^2 - 8*(B*b^3*d*e^2 - (7*B*a*b^2 - 6*A*b^3)*e^3)*x^3 - 4*(7*B*b^3*d^2*e - 2*(25*B*a*b^2 - 21*A*b^3)*d*e
^2 + (7*B*a^2*b - 6*A*a*b^2)*e^3)*x^2 - (35*B*b^3*d^3 - 7*(37*B*a*b^2 - 30*A*b^3)*d^2*e + (101*B*a^2*b - 84*A*
a*b^2)*d*e^2 - 3*(7*B*a^3 - 6*A*a^2*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^8 - 4*a*b^3*d^7*e + 6*a^2*b^
2*d^6*e^2 - 4*a^3*b*d^5*e^3 + a^4*d^4*e^4 + (b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7
+ a^4*e^8)*x^4 + 4*(b^4*d^5*e^3 - 4*a*b^3*d^4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d^2*e^6 + a^4*d*e^7)*x^3 + 6*
(b^4*d^6*e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*d^2*e^6)*x^2 + 4*(b^4*d^7*e - 4*a*b
^3*d^6*e^2 + 6*a^2*b^2*d^5*e^3 - 4*a^3*b*d^4*e^4 + a^4*d^3*e^5)*x)

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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((B*x+A)/(e*x+d)**(9/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.21668, size = 782, normalized size = 3.95 \begin{align*} -\frac{{\left ({\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{8} d{\left | b \right |} e^{5} - 7 \, B a b^{7}{\left | b \right |} e^{6} + 6 \, A b^{8}{\left | b \right |} e^{6}\right )}{\left (b x + a\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}} + \frac{7 \,{\left (B b^{9} d^{2}{\left | b \right |} e^{4} - 8 \, B a b^{8} d{\left | b \right |} e^{5} + 6 \, A b^{9} d{\left | b \right |} e^{5} + 7 \, B a^{2} b^{7}{\left | b \right |} e^{6} - 6 \, A a b^{8}{\left | b \right |} e^{6}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )} + \frac{35 \,{\left (B b^{10} d^{3}{\left | b \right |} e^{3} - 9 \, B a b^{9} d^{2}{\left | b \right |} e^{4} + 6 \, A b^{10} d^{2}{\left | b \right |} e^{4} + 15 \, B a^{2} b^{8} d{\left | b \right |} e^{5} - 12 \, A a b^{9} d{\left | b \right |} e^{5} - 7 \, B a^{3} b^{7}{\left | b \right |} e^{6} + 6 \, A a^{2} b^{8}{\left | b \right |} e^{6}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )}{\left (b x + a\right )} - \frac{105 \,{\left (B a b^{10} d^{3}{\left | b \right |} e^{3} - A b^{11} d^{3}{\left | b \right |} e^{3} - 3 \, B a^{2} b^{9} d^{2}{\left | b \right |} e^{4} + 3 \, A a b^{10} d^{2}{\left | b \right |} e^{4} + 3 \, B a^{3} b^{8} d{\left | b \right |} e^{5} - 3 \, A a^{2} b^{9} d{\left | b \right |} e^{5} - B a^{4} b^{7}{\left | b \right |} e^{6} + A a^{3} b^{8}{\left | b \right |} e^{6}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )} \sqrt{b x + a}}{80640 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/80640*((4*(b*x + a)*(2*(B*b^8*d*abs(b)*e^5 - 7*B*a*b^7*abs(b)*e^6 + 6*A*b^8*abs(b)*e^6)*(b*x + a)/(b^16*d^4
*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12) + 7*(B*b^9*d^2*abs(b)*e^4 -
8*B*a*b^8*d*abs(b)*e^5 + 6*A*b^9*d*abs(b)*e^5 + 7*B*a^2*b^7*abs(b)*e^6 - 6*A*a*b^8*abs(b)*e^6)/(b^16*d^4*e^8
- 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12)) + 35*(B*b^10*d^3*abs(b)*e^3 - 9
*B*a*b^9*d^2*abs(b)*e^4 + 6*A*b^10*d^2*abs(b)*e^4 + 15*B*a^2*b^8*d*abs(b)*e^5 - 12*A*a*b^9*d*abs(b)*e^5 - 7*B*
a^3*b^7*abs(b)*e^6 + 6*A*a^2*b^8*abs(b)*e^6)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^
13*d*e^11 + a^4*b^12*e^12))*(b*x + a) - 105*(B*a*b^10*d^3*abs(b)*e^3 - A*b^11*d^3*abs(b)*e^3 - 3*B*a^2*b^9*d^2
*abs(b)*e^4 + 3*A*a*b^10*d^2*abs(b)*e^4 + 3*B*a^3*b^8*d*abs(b)*e^5 - 3*A*a^2*b^9*d*abs(b)*e^5 - B*a^4*b^7*abs(
b)*e^6 + A*a^3*b^8*abs(b)*e^6)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^
4*b^12*e^12))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)