3.2230 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\)
Optimal. Leaf size=167 \[ -\frac{2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}+\frac{2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}} \]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) - (2*B*(a + b*x)^(5/2))/(5*e^2*(d + e*x)^(5
/2)) - (2*b*B*(a + b*x)^(3/2))/(3*e^3*(d + e*x)^(3/2)) - (2*b^2*B*Sqrt[a + b*x])/(e^4*Sqrt[d + e*x]) + (2*b^(5
/2)*B*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/e^(9/2)
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Rubi [A] time = 0.0963402, antiderivative size = 167, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used =
{78, 47, 63, 217, 206} \[ -\frac{2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}+\frac{2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) - (2*B*(a + b*x)^(5/2))/(5*e^2*(d + e*x)^(5
/2)) - (2*b*B*(a + b*x)^(3/2))/(3*e^3*(d + e*x)^(3/2)) - (2*b^2*B*Sqrt[a + b*x])/(e^4*Sqrt[d + e*x]) + (2*b^(5
/2)*B*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/e^(9/2)
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 47
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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{(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{B \int \frac{(a+b x)^{5/2}}{(d+e x)^{7/2}} \, dx}{e}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{(b B) \int \frac{(a+b x)^{3/2}}{(d+e x)^{5/2}} \, dx}{e^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}+\frac{\left (b^2 B\right ) \int \frac{\sqrt{a+b x}}{(d+e x)^{3/2}} \, dx}{e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}+\frac{\left (b^3 B\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{e^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}+\frac{\left (2 b^2 B\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{e^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}+\frac{\left (2 b^2 B\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{e^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}+\frac{2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}\\ \end{align*}
Mathematica [A] time = 2.19471, size = 222, normalized size = 1.33 \[ \frac{2 \left (e^4 (a+b x)^4 (A e-B d)-7 b^2 B e (a+b x) (d+e x)^3 (b d-a e)-\frac{7}{3} b B e^2 (a+b x)^2 (d+e x)^2 (b d-a e)+\frac{7}{5} B e^3 (a+b x)^3 (d+e x) (a e-b d)+\frac{7 B \sqrt{e} \sqrt{a+b x} (b d-a e)^{9/2} \left (\frac{b (d+e x)}{b d-a e}\right )^{7/2} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{b}\right )}{7 e^5 \sqrt{a+b x} (d+e x)^{7/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]
[Out]
(2*(e^4*(-(B*d) + A*e)*(a + b*x)^4 + (7*B*e^3*(-(b*d) + a*e)*(a + b*x)^3*(d + e*x))/5 - (7*b*B*e^2*(b*d - a*e)
*(a + b*x)^2*(d + e*x)^2)/3 - 7*b^2*B*e*(b*d - a*e)*(a + b*x)*(d + e*x)^3 + (7*B*Sqrt[e]*(b*d - a*e)^(9/2)*Sqr
t[a + b*x]*((b*(d + e*x))/(b*d - a*e))^(7/2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/b))/(7*e^5*(b*d
- a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2))
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Maple [B] time = 0.025, size = 1089, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x)
[Out]
-1/105*(b*x+a)^(1/2)*(154*B*x^2*a^2*b*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-812*B*x^2*b^3*d^2*e^2*((b*x+a)*(
e*x+d))^(1/2)*(b*e)^(1/2)+90*A*x*a^2*b*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-700*B*x*b^3*d^3*e*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)+28*B*a^2*b*d^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+140*B*a*b^2*d^3*e*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2)+420*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*b^4*
d^4*e+30*A*a^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-210*B*b^3*d^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+322*B
*x^3*a*b^2*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-352*B*x^3*b^3*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*
A*x^2*a*b^2*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d^4*e+30*A*x^3*b^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+105*B*ln(1/2*(2*b*x*e
+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^5+476*B*x*a*b^2*d^2*e^2*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)+568*B*x^2*a*b^2*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+92*B*x*a^2*b*d*e^3*((b*x+a)*(e*x+d)
)^(1/2)*(b*e)^(1/2)-420*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^3*a*b^
3*d*e^4-630*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*a*b^3*d^2*e^3-42
0*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^3*d^3*e^2+42*B*x*a^3*e^4
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12*B*a^3*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-105*B*ln(1/2*(2*b*x*e+
2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^4*a*b^3*e^5+105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^4*b^4*d*e^4+420*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^3*b^4*d^2*e^3+630*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*
e+b*d)/(b*e)^(1/2))*x^2*b^4*d^3*e^2)/((b*x+a)*(e*x+d))^(1/2)/(a*e-b*d)/(b*e)^(1/2)/(e*x+d)^(7/2)/e^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)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 98.2997, size = 2225, normalized size = 13.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="fricas")
[Out]
[1/210*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x
^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(b/e)*log(8*b^2*e^2*x^
2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x + b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(b/e) + 8*(b
^2*d*e + a*b*e^2)*x) - 4*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4
+ (176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b
+ 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)
*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 - a*e^9)*x^4 + 4*(b*d^2*e^7 - a*d*e^8)
*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b*d^4*e^5 - a*d^3*e^6)*x), -1/105*(105*(B*b^3*d^5 - B*a*b^2*d^4*e +
(B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^3)*
x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(-b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x
+ d)*sqrt(-b/e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) + 2*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^
2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + (176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e
^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B*a*b^2*d^2*e^2 - 46*B*a^
2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 -
a*e^9)*x^4 + 4*(b*d^2*e^7 - a*d*e^8)*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b*d^4*e^5 - a*d^3*e^6)*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)**(5/2)*(B*x+A)/(e*x+d)**(9/2),x)
[Out]
Timed out
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Giac [B] time = 2.9056, size = 946, normalized size = 5.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")
[Out]
1/768*B*sqrt(b)*abs(b)*e^(1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/
(b^10*d*e^6 - a*b^9*e^7) + 1/80640*(((b*x + a)*((176*B*b^10*d^3*abs(b)*e^6 - 513*B*a*b^9*d^2*abs(b)*e^7 - 15*A
*b^10*d^2*abs(b)*e^7 + 498*B*a^2*b^8*d*abs(b)*e^8 + 30*A*a*b^9*d*abs(b)*e^8 - 161*B*a^3*b^7*abs(b)*e^9 - 15*A*
a^2*b^8*abs(b)*e^9)*(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) + 406*(B*b^11*d^4*abs(b)*e^5 - 4*B*a*b^10*d^3*abs(b)*e^6 + 6*B*a^2*b^9*d^2*abs(b)*e^7 - 4*B*a^3*b^
8*d*abs(b)*e^8 + B*a^4*b^7*abs(b)*e^9)/(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)) + 350*(B*b^12*d^5*abs(b)*e^4 - 5*B*a*b^11*d^4*abs(b)*e^5 + 10*B*a^2*b^10*d^3*abs(b)*e^6
- 10*B*a^3*b^9*d^2*abs(b)*e^7 + 5*B*a^4*b^8*d*abs(b)*e^8 - B*a^5*b^7*abs(b)*e^9)/(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*b^13*d^6*abs(b)*e^3 - 6*B*a
*b^12*d^5*abs(b)*e^4 + 15*B*a^2*b^11*d^4*abs(b)*e^5 - 20*B*a^3*b^10*d^3*abs(b)*e^6 + 15*B*a^4*b^9*d^2*abs(b)*e
^7 - 6*B*a^5*b^8*d*abs(b)*e^8 + B*a^6*b^7*abs(b)*e^9)/(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)