3.23.53 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx\) [2253]

Optimal. Leaf size=302 \[ \frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}} \]

[Out]

-2/3*(A*b-B*a)*(e*x+d)^(9/2)/b/(-a*e+b*d)/(b*x+a)^(3/2)+35/8*(-a*e+b*d)^2*(2*A*b*e-3*B*a*e+B*b*d)*arctanh(e^(1
/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))*e^(1/2)/b^(11/2)-2*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*
d)/(b*x+a)^(1/2)+35/12*e*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(3/2)*(b*x+a)^(1/2)/b^4+7/3*e*(2*A*b*e-3*B*a*e+B*b*d)
*(e*x+d)^(5/2)*(b*x+a)^(1/2)/b^3/(-a*e+b*d)+35/8*e*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1
/2)/b^5

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 52, 65, 223, 212} \begin {gather*} \frac {35 \sqrt {e} (b d-a e)^2 (-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}}+\frac {35 e \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^5}+\frac {35 e \sqrt {a+b x} (d+e x)^{3/2} (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac {7 e \sqrt {a+b x} (d+e x)^{5/2} (-3 a B e+2 A b e+b B d)}{3 b^3 (b d-a e)}-\frac {2 (d+e x)^{7/2} (-3 a B e+2 A b e+b B d)}{b^2 \sqrt {a+b x} (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*e*(b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^5) + (35*e*(b*B*d + 2*A*b*e -
3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^4) + (7*e*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5
/2))/(3*b^3*(b*d - a*e)) - (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(b^2*(b*d - a*e)*Sqrt[a + b*x]) - (
2*(A*b - a*B)*(d + e*x)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (35*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2*A*b*e
- 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(11/2))

Rule 49

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 52

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

Rule 65

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 79

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] || L
tQ[p, n]))))

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 223

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]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(b B d+2 A b e-3 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^{3/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(7 e (b B d+2 A b e-3 a B e)) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}} \, dx}{b^2 (b d-a e)}\\ &=\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(35 e (b B d+2 A b e-3 a B e)) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^3}\\ &=\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(35 e (b d-a e) (b B d+2 A b e-3 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{8 b^4}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\left (35 e (b d-a e)^2 (b B d+2 A b e-3 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{16 b^5}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\left (35 e (b d-a e)^2 (b B d+2 A b e-3 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^6}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {\left (35 e (b d-a e)^2 (b B d+2 A b e-3 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{8 b^6}\\ &=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.84, size = 313, normalized size = 1.04 \begin {gather*} \frac {\sqrt {d+e x} \left (-2 A b \left (105 a^3 e^3+35 a^2 b e^2 (-5 d+4 e x)+7 a b^2 e \left (8 d^2-34 d e x+3 e^2 x^2\right )+b^3 \left (8 d^3+80 d^2 e x-39 d e^2 x^2-6 e^3 x^3\right )\right )+B \left (315 a^4 e^3+210 a^3 b e^2 (-3 d+2 e x)+7 a^2 b^2 e \left (49 d^2-122 d e x+9 e^2 x^2\right )+b^4 x \left (-48 d^3+87 d^2 e x+38 d e^2 x^2+8 e^3 x^3\right )-2 a b^3 \left (16 d^3-239 d^2 e x+69 d e^2 x^2+9 e^3 x^3\right )\right )\right )}{24 b^5 (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-2*A*b*(105*a^3*e^3 + 35*a^2*b*e^2*(-5*d + 4*e*x) + 7*a*b^2*e*(8*d^2 - 34*d*e*x + 3*e^2*x^2) +
b^3*(8*d^3 + 80*d^2*e*x - 39*d*e^2*x^2 - 6*e^3*x^3)) + B*(315*a^4*e^3 + 210*a^3*b*e^2*(-3*d + 2*e*x) + 7*a^2*
b^2*e*(49*d^2 - 122*d*e*x + 9*e^2*x^2) + b^4*x*(-48*d^3 + 87*d^2*e*x + 38*d*e^2*x^2 + 8*e^3*x^3) - 2*a*b^3*(16
*d^3 - 239*d^2*e*x + 69*d*e^2*x^2 + 9*e^3*x^3))))/(24*b^5*(a + b*x)^(3/2)) + (35*Sqrt[e]*(b*d - a*e)^2*(b*B*d
+ 2*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*b^(11/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1881\) vs. \(2(260)=520\).
time = 0.12, size = 1882, normalized size = 6.23

method	result	size
default	\(\text {Expression too large to display}\)	\(1882\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(e*x+d)^(1/2)*(-315*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2
*e^4*x^2+105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^3*e*x^2+420*A
*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*e^4*x-630*B*ln(1/2*(2*b*e
*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*e^4*x-420*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d*e^3+210*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^2*e^2+735*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a
*e+b*d)/(b*e)^(1/2))*a^4*b*d*e^3-525*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1
/2))*a^3*b^2*d^2*e^2+105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3
*d^3*e+24*A*b^4*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*b^4*d^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
-420*A*a^3*b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-64*B*a*b^3*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*A*
ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*e^4*x^2+210*A*ln(1/2*(2*b*
e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^2*e^2*x^2+16*B*b^4*e^3*x^4*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)-1708*B*a^2*b^2*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+956*B*a*b^3*d^2*e*x*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)-315*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*
a^5*e^4-840*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d*e^3*x+420*
A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^2*e^2*x+1470*B*ln(1/2*(2
*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d*e^3*x-1050*B*ln(1/2*(2*b*e*x+2*((
b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^2*e^2*x+210*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^3*e*x-36*B*a*b^3*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)+76*B*b^4*d*e^2*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-84*A*a*b^3*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)+156*A*b^4*d*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+126*B*a^2*b^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)+174*B*b^4*d^2*e*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-560*A*a^2*b^2*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)-320*A*b^4*d^2*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+840*B*a^3*b*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)+700*A*a^2*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-224*A*a*b^3*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)-1260*B*a^3*b*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+686*B*a^2*b^2*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)-420*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d*e^3*x^2+73
5*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d*e^3*x^2-525*B*ln(1/2
*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^2*e^2*x^2-276*B*a*b^3*d*e^2*x^2*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+952*A*a*b^3*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-32*A*b^4*d^3*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+630*B*a^4*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*A*ln(1/2*(2*b*e*x+2*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*e^4)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(3
/2)/b^5

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (282) = 564\).
time = 2.77, size = 1234, normalized size = 4.09 \begin {gather*} \left [-\frac {\frac {105 \, {\left (B b^{5} d^{3} x^{2} + 2 \, B a b^{4} d^{3} x + B a^{2} b^{3} d^{3} - {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} e^{3} + {\left ({\left (7 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d x^{2} + 2 \, {\left (7 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} d x + {\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} d\right )} e^{2} - {\left ({\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} d^{2} x^{2} + 2 \, {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} x + {\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2}\right )} e\right )} e^{\frac {1}{2}} \log \left (b^{2} d^{2} - \frac {4 \, {\left (b^{2} d + {\left (2 \, b^{2} x + a b\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} e^{\frac {1}{2}}}{\sqrt {b}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right )}{\sqrt {b}} + 4 \, {\left (48 \, B b^{4} d^{3} x + 16 \, {\left (2 \, B a b^{3} + A b^{4}\right )} d^{3} - {\left (8 \, B b^{4} x^{4} + 315 \, B a^{4} - 210 \, A a^{3} b - 6 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} e^{3} - 2 \, {\left (19 \, B b^{4} d x^{3} - 3 \, {\left (23 \, B a b^{3} - 13 \, A b^{4}\right )} d x^{2} - 7 \, {\left (61 \, B a^{2} b^{2} - 34 \, A a b^{3}\right )} d x - 35 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d\right )} e^{2} - {\left (87 \, B b^{4} d^{2} x^{2} + 2 \, {\left (239 \, B a b^{3} - 80 \, A b^{4}\right )} d^{2} x + 7 \, {\left (49 \, B a^{2} b^{2} - 16 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{96 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {105 \, {\left (B b^{5} d^{3} x^{2} + 2 \, B a b^{4} d^{3} x + B a^{2} b^{3} d^{3} - {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} e^{3} + {\left ({\left (7 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d x^{2} + 2 \, {\left (7 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} d x + {\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} d\right )} e^{2} - {\left ({\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} d^{2} x^{2} + 2 \, {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} x + {\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2}\right )} e\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {-\frac {e}{b}}}{2 \, {\left ({\left (b x^{2} + a x\right )} e^{2} + {\left (b d x + a d\right )} e\right )}}\right ) + 2 \, {\left (48 \, B b^{4} d^{3} x + 16 \, {\left (2 \, B a b^{3} + A b^{4}\right )} d^{3} - {\left (8 \, B b^{4} x^{4} + 315 \, B a^{4} - 210 \, A a^{3} b - 6 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} e^{3} - 2 \, {\left (19 \, B b^{4} d x^{3} - 3 \, {\left (23 \, B a b^{3} - 13 \, A b^{4}\right )} d x^{2} - 7 \, {\left (61 \, B a^{2} b^{2} - 34 \, A a b^{3}\right )} d x - 35 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d\right )} e^{2} - {\left (87 \, B b^{4} d^{2} x^{2} + 2 \, {\left (239 \, B a b^{3} - 80 \, A b^{4}\right )} d^{2} x + 7 \, {\left (49 \, B a^{2} b^{2} - 16 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{48 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(105*(B*b^5*d^3*x^2 + 2*B*a*b^4*d^3*x + B*a^2*b^3*d^3 - (3*B*a^5 - 2*A*a^4*b + (3*B*a^3*b^2 - 2*A*a^2*b
^3)*x^2 + 2*(3*B*a^4*b - 2*A*a^3*b^2)*x)*e^3 + ((7*B*a^2*b^3 - 4*A*a*b^4)*d*x^2 + 2*(7*B*a^3*b^2 - 4*A*a^2*b^3
)*d*x + (7*B*a^4*b - 4*A*a^3*b^2)*d)*e^2 - ((5*B*a*b^4 - 2*A*b^5)*d^2*x^2 + 2*(5*B*a^2*b^3 - 2*A*a*b^4)*d^2*x
+ (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2)*e)*e^(1/2)*log(b^2*d^2 - 4*(b^2*d + (2*b^2*x + a*b)*e)*sqrt(b*x + a)*sqrt(x
*e + d)*e^(1/2)/sqrt(b) + (8*b^2*x^2 + 8*a*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e)/sqrt(b) + 4*(48*B*b^4*d
^3*x + 16*(2*B*a*b^3 + A*b^4)*d^3 - (8*B*b^4*x^4 + 315*B*a^4 - 210*A*a^3*b - 6*(3*B*a*b^3 - 2*A*b^4)*x^3 + 21*
(3*B*a^2*b^2 - 2*A*a*b^3)*x^2 + 140*(3*B*a^3*b - 2*A*a^2*b^2)*x)*e^3 - 2*(19*B*b^4*d*x^3 - 3*(23*B*a*b^3 - 13*
A*b^4)*d*x^2 - 7*(61*B*a^2*b^2 - 34*A*a*b^3)*d*x - 35*(9*B*a^3*b - 5*A*a^2*b^2)*d)*e^2 - (87*B*b^4*d^2*x^2 + 2
*(239*B*a*b^3 - 80*A*b^4)*d^2*x + 7*(49*B*a^2*b^2 - 16*A*a*b^3)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(b^7*x^2
+ 2*a*b^6*x + a^2*b^5), -1/48*(105*(B*b^5*d^3*x^2 + 2*B*a*b^4*d^3*x + B*a^2*b^3*d^3 - (3*B*a^5 - 2*A*a^4*b + (
3*B*a^3*b^2 - 2*A*a^2*b^3)*x^2 + 2*(3*B*a^4*b - 2*A*a^3*b^2)*x)*e^3 + ((7*B*a^2*b^3 - 4*A*a*b^4)*d*x^2 + 2*(7*
B*a^3*b^2 - 4*A*a^2*b^3)*d*x + (7*B*a^4*b - 4*A*a^3*b^2)*d)*e^2 - ((5*B*a*b^4 - 2*A*b^5)*d^2*x^2 + 2*(5*B*a^2*
b^3 - 2*A*a*b^4)*d^2*x + (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2)*e)*sqrt(-e/b)*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(
b*x + a)*sqrt(x*e + d)*sqrt(-e/b)/((b*x^2 + a*x)*e^2 + (b*d*x + a*d)*e)) + 2*(48*B*b^4*d^3*x + 16*(2*B*a*b^3 +
A*b^4)*d^3 - (8*B*b^4*x^4 + 315*B*a^4 - 210*A*a^3*b - 6*(3*B*a*b^3 - 2*A*b^4)*x^3 + 21*(3*B*a^2*b^2 - 2*A*a*b
^3)*x^2 + 140*(3*B*a^3*b - 2*A*a^2*b^2)*x)*e^3 - 2*(19*B*b^4*d*x^3 - 3*(23*B*a*b^3 - 13*A*b^4)*d*x^2 - 7*(61*B
*a^2*b^2 - 34*A*a*b^3)*d*x - 35*(9*B*a^3*b - 5*A*a^2*b^2)*d)*e^2 - (87*B*b^4*d^2*x^2 + 2*(239*B*a*b^3 - 80*A*b
^4)*d^2*x + 7*(49*B*a^2*b^2 - 16*A*a*b^3)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)
]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1654 vs. \(2 (282) = 564\).
time = 1.08, size = 1654, normalized size = 5.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs(b)*e^3/b^7 + (19*B*b^21
*d*abs(b)*e^6 - 25*B*a*b^20*abs(b)*e^7 + 6*A*b^21*abs(b)*e^7)*e^(-4)/b^27) + 3*(29*B*b^22*d^2*abs(b)*e^5 - 84*
B*a*b^21*d*abs(b)*e^6 + 26*A*b^22*d*abs(b)*e^6 + 55*B*a^2*b^20*abs(b)*e^7 - 26*A*a*b^21*abs(b)*e^7)*e^(-4)/b^2
7) - 35/16*(B*b^(7/2)*d^3*abs(b)*e^(1/2) - 5*B*a*b^(5/2)*d^2*abs(b)*e^(3/2) + 2*A*b^(7/2)*d^2*abs(b)*e^(3/2) +
7*B*a^2*b^(3/2)*d*abs(b)*e^(5/2) - 4*A*a*b^(5/2)*d*abs(b)*e^(5/2) - 3*B*a^3*sqrt(b)*abs(b)*e^(7/2) + 2*A*a^2*
b^(3/2)*abs(b)*e^(7/2))*log((sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/b^7 - 4/3
*(3*B*b^(17/2)*d^6*abs(b)*e^(1/2) - 28*B*a*b^(15/2)*d^5*abs(b)*e^(3/2) + 10*A*b^(17/2)*d^5*abs(b)*e^(3/2) - 6*
(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^(13/2)*d^5*abs(b)*e^(1/2) + 95*B*a
^2*b^(13/2)*d^4*abs(b)*e^(5/2) - 50*A*a*b^(15/2)*d^4*abs(b)*e^(5/2) + 48*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt
(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^(11/2)*d^4*abs(b)*e^(3/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^2*A*b^(13/2)*d^4*abs(b)*e^(3/2) + 3*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*
d + (b*x + a)*b*e - a*b*e))^4*B*b^(9/2)*d^4*abs(b)*e^(1/2) - 160*B*a^3*b^(11/2)*d^3*abs(b)*e^(7/2) + 100*A*a^2
*b^(13/2)*d^3*abs(b)*e^(7/2) - 132*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a
^2*b^(9/2)*d^3*abs(b)*e^(5/2) + 72*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a
*b^(11/2)*d^3*abs(b)*e^(5/2) - 24*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a*
b^(7/2)*d^3*abs(b)*e^(3/2) + 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*b^(9
/2)*d^3*abs(b)*e^(3/2) + 145*B*a^4*b^(9/2)*d^2*abs(b)*e^(9/2) - 100*A*a^3*b^(11/2)*d^2*abs(b)*e^(9/2) + 168*(s
qrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^3*b^(7/2)*d^2*abs(b)*e^(7/2) - 108*(
sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^2*b^(9/2)*d^2*abs(b)*e^(7/2) + 54*(
sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^2*b^(5/2)*d^2*abs(b)*e^(5/2) - 36*(
sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*a*b^(7/2)*d^2*abs(b)*e^(5/2) - 68*B*a
^5*b^(7/2)*d*abs(b)*e^(11/2) + 50*A*a^4*b^(9/2)*d*abs(b)*e^(11/2) - 102*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(
b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^4*b^(5/2)*d*abs(b)*e^(9/2) + 72*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^
2*d + (b*x + a)*b*e - a*b*e))^2*A*a^3*b^(7/2)*d*abs(b)*e^(9/2) - 48*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*
d + (b*x + a)*b*e - a*b*e))^4*B*a^3*b^(3/2)*d*abs(b)*e^(7/2) + 36*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e))^4*A*a^2*b^(5/2)*d*abs(b)*e^(7/2) + 13*B*a^6*b^(5/2)*abs(b)*e^(13/2) - 10*A*a^5*b^(7/
2)*abs(b)*e^(13/2) + 24*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a^5*b^(3/2)*
abs(b)*e^(11/2) - 18*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*A*a^4*b^(5/2)*abs
(b)*e^(11/2) + 15*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*B*a^4*sqrt(b)*abs(b)
*e^(9/2) - 12*(sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^4*A*a^3*b^(3/2)*abs(b)*e^(
9/2))/((b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^(1/2) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)^3*b^6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2), x)

________________________________________________________________________________________