3.1751 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=365 \[ -\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{6 e (d+e x)^6 (b d-a e)}+\frac {10 b^2 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac {B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}+\frac {5 b^4 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac {5 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^2} \]

[Out]

-1/6*(-A*e+B*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)/(e*x+d)^6+1/5*B*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b
*x+a)/(e*x+d)^5-5/4*b*B*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^4+10/3*b^2*B*(-a*e+b*d)^3*((b*x+a)^
2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3-5*b^3*B*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^2+5*b^4*B*(-a*e+b*d)
*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+b^5*B*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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Rubi [A]  time = 0.29, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 78, 43} \[ -\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2} (B d-A e)}{6 e (d+e x)^6 (b d-a e)}+\frac {5 b^4 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac {5 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)^2}+\frac {10 b^2 B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac {B \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e*(b*d - a*e)*(d + e*x)^6) + (B*(b*d - a*e)^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^5) - (5*b*B*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(4*e^7*(a + b*x)*(d + e*x)^4) + (10*b^2*B*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d +
e*x)^3) - (5*b^3*B*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^2) + (5*b^4*B*(b*d -
a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e
*x])/(e^7*(a + b*x))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^7} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {\left (B \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^6} \, dx}{b^4 e \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {\left (B \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^6}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^5}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^4}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^3}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^2}+\frac {b^{10}}{e^5 (d+e x)}\right ) \, dx}{b^4 e \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e (b d-a e) (d+e x)^6}+\frac {B (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}-\frac {5 b B (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {10 b^2 B (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b^3 B (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}+\frac {5 b^4 B (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 477, normalized size = 1.31 \[ -\frac {\sqrt {(a+b x)^2} \left (2 a^5 e^5 (5 A e+B (d+6 e x))+5 a^4 b e^4 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+10 a b^4 e \left (A e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+b^5 \left (10 A e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )-B d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )-60 b^5 B (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(Sqrt[(a + b*x)^2]*(2*a^5*e^5*(5*A*e + B*(d + 6*e*x)) + 5*a^4*b*e^4*(2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*
x + 15*e^2*x^2)) + 10*a^3*b^2*e^3*(A*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e
^3*x^3)) + 10*a^2*b^3*e^2*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 15*d^2*e
^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 10*a*b^4*e*(A*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e
^4*x^4) + 5*B*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) + b^5*(10*A*e*(d
^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - B*d*(147*d^5 + 822*d^4*e*x + 18
75*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) - 60*b^5*B*(d + e*x)^6*Log[d + e*x]))/(e^7*
(a + b*x)*(d + e*x)^6)

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fricas [B]  time = 0.89, size = 703, normalized size = 1.93 \[ \frac {147 \, B b^{5} d^{6} - 10 \, A a^{5} e^{6} - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 60 \, {\left (6 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 150 \, {\left (9 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} - {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 200 \, {\left (11 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} - {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 75 \, {\left (25 \, B b^{5} d^{4} e^{2} - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} - 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} - {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, B b^{5} d^{5} e - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} - 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \, {\left (B b^{5} e^{6} x^{6} + 6 \, B b^{5} d e^{5} x^{5} + 15 \, B b^{5} d^{2} e^{4} x^{4} + 20 \, B b^{5} d^{3} e^{3} x^{3} + 15 \, B b^{5} d^{4} e^{2} x^{2} + 6 \, B b^{5} d^{5} e x + B b^{5} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(147*B*b^5*d^6 - 10*A*a^5*e^6 - 10*(5*B*a*b^4 + A*b^5)*d^5*e - 10*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 10*(B
*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 5*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 2*(B*a^5 + 5*A*a^4*b)*d*e^5 + 60*(6*B*b^5*
d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 150*(9*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 - (2*B*a^2*b^3 + A*a*b
^4)*e^6)*x^4 + 200*(11*B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 - (2*B*a^2*b^3 + A*a*b^4)*d*e^5 - (B*a^3*b^
2 + A*a^2*b^3)*e^6)*x^3 + 75*(25*B*b^5*d^4*e^2 - 2*(5*B*a*b^4 + A*b^5)*d^3*e^3 - 2*(2*B*a^2*b^3 + A*a*b^4)*d^2
*e^4 - 2*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 - (B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 6*(137*B*b^5*d^5*e - 10*(5*B*a*b^4
+ A*b^5)*d^4*e^2 - 10*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 10*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 - 5*(B*a^4*b + 2*A
*a^3*b^2)*d*e^5 - 2*(B*a^5 + 5*A*a^4*b)*e^6)*x + 60*(B*b^5*e^6*x^6 + 6*B*b^5*d*e^5*x^5 + 15*B*b^5*d^2*e^4*x^4
+ 20*B*b^5*d^3*e^3*x^3 + 15*B*b^5*d^4*e^2*x^2 + 6*B*b^5*d^5*e*x + B*b^5*d^6)*log(e*x + d))/(e^13*x^6 + 6*d*e^1
2*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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giac [B]  time = 0.30, size = 871, normalized size = 2.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b^5*e^(-7)*log(abs(x*e + d))*sgn(b*x + a) + 1/60*(60*(6*B*b^5*d*e^4*sgn(b*x + a) - 5*B*a*b^4*e^5*sgn(b*x + a
) - A*b^5*e^5*sgn(b*x + a))*x^5 + 150*(9*B*b^5*d^2*e^3*sgn(b*x + a) - 5*B*a*b^4*d*e^4*sgn(b*x + a) - A*b^5*d*e
^4*sgn(b*x + a) - 2*B*a^2*b^3*e^5*sgn(b*x + a) - A*a*b^4*e^5*sgn(b*x + a))*x^4 + 200*(11*B*b^5*d^3*e^2*sgn(b*x
+ a) - 5*B*a*b^4*d^2*e^3*sgn(b*x + a) - A*b^5*d^2*e^3*sgn(b*x + a) - 2*B*a^2*b^3*d*e^4*sgn(b*x + a) - A*a*b^4
*d*e^4*sgn(b*x + a) - B*a^3*b^2*e^5*sgn(b*x + a) - A*a^2*b^3*e^5*sgn(b*x + a))*x^3 + 75*(25*B*b^5*d^4*e*sgn(b*
x + a) - 10*B*a*b^4*d^3*e^2*sgn(b*x + a) - 2*A*b^5*d^3*e^2*sgn(b*x + a) - 4*B*a^2*b^3*d^2*e^3*sgn(b*x + a) - 2
*A*a*b^4*d^2*e^3*sgn(b*x + a) - 2*B*a^3*b^2*d*e^4*sgn(b*x + a) - 2*A*a^2*b^3*d*e^4*sgn(b*x + a) - B*a^4*b*e^5*
sgn(b*x + a) - 2*A*a^3*b^2*e^5*sgn(b*x + a))*x^2 + 6*(137*B*b^5*d^5*sgn(b*x + a) - 50*B*a*b^4*d^4*e*sgn(b*x +
a) - 10*A*b^5*d^4*e*sgn(b*x + a) - 20*B*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*A*a*b^4*d^3*e^2*sgn(b*x + a) - 10*B*
a^3*b^2*d^2*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^2*e^3*sgn(b*x + a) - 5*B*a^4*b*d*e^4*sgn(b*x + a) - 10*A*a^3*b^2
*d*e^4*sgn(b*x + a) - 2*B*a^5*e^5*sgn(b*x + a) - 10*A*a^4*b*e^5*sgn(b*x + a))*x + (147*B*b^5*d^6*sgn(b*x + a)
- 50*B*a*b^4*d^5*e*sgn(b*x + a) - 10*A*b^5*d^5*e*sgn(b*x + a) - 20*B*a^2*b^3*d^4*e^2*sgn(b*x + a) - 10*A*a*b^4
*d^4*e^2*sgn(b*x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x + a) - 5*B*a^4*b*d^2*
e^4*sgn(b*x + a) - 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - 2*B*a^5*d*e^5*sgn(b*x + a) - 10*A*a^4*b*d*e^5*sgn(b*x +
a) - 10*A*a^5*e^6*sgn(b*x + a))*e^(-1))*e^(-6)/(x*e + d)^6

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maple [B]  time = 0.07, size = 809, normalized size = 2.22 \[ -\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-60 B \,b^{5} e^{6} x^{6} \ln \left (e x +d \right )-360 B \,b^{5} d \,e^{5} x^{5} \ln \left (e x +d \right )+60 A \,b^{5} e^{6} x^{5}+300 B a \,b^{4} e^{6} x^{5}-900 B \,b^{5} d^{2} e^{4} x^{4} \ln \left (e x +d \right )-360 B \,b^{5} d \,e^{5} x^{5}+150 A a \,b^{4} e^{6} x^{4}+150 A \,b^{5} d \,e^{5} x^{4}+300 B \,a^{2} b^{3} e^{6} x^{4}+750 B a \,b^{4} d \,e^{5} x^{4}-1200 B \,b^{5} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-1350 B \,b^{5} d^{2} e^{4} x^{4}+200 A \,a^{2} b^{3} e^{6} x^{3}+200 A a \,b^{4} d \,e^{5} x^{3}+200 A \,b^{5} d^{2} e^{4} x^{3}+200 B \,a^{3} b^{2} e^{6} x^{3}+400 B \,a^{2} b^{3} d \,e^{5} x^{3}+1000 B a \,b^{4} d^{2} e^{4} x^{3}-900 B \,b^{5} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-2200 B \,b^{5} d^{3} e^{3} x^{3}+150 A \,a^{3} b^{2} e^{6} x^{2}+150 A \,a^{2} b^{3} d \,e^{5} x^{2}+150 A a \,b^{4} d^{2} e^{4} x^{2}+150 A \,b^{5} d^{3} e^{3} x^{2}+75 B \,a^{4} b \,e^{6} x^{2}+150 B \,a^{3} b^{2} d \,e^{5} x^{2}+300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+750 B a \,b^{4} d^{3} e^{3} x^{2}-360 B \,b^{5} d^{5} e x \ln \left (e x +d \right )-1875 B \,b^{5} d^{4} e^{2} x^{2}+60 A \,a^{4} b \,e^{6} x +60 A \,a^{3} b^{2} d \,e^{5} x +60 A \,a^{2} b^{3} d^{2} e^{4} x +60 A a \,b^{4} d^{3} e^{3} x +60 A \,b^{5} d^{4} e^{2} x +12 B \,a^{5} e^{6} x +30 B \,a^{4} b d \,e^{5} x +60 B \,a^{3} b^{2} d^{2} e^{4} x +120 B \,a^{2} b^{3} d^{3} e^{3} x +300 B a \,b^{4} d^{4} e^{2} x -60 B \,b^{5} d^{6} \ln \left (e x +d \right )-822 B \,b^{5} d^{5} e x +10 A \,a^{5} e^{6}+10 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+10 A \,b^{5} d^{5} e +2 B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+50 B a \,b^{4} d^{5} e -147 B \,b^{5} d^{6}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right )^{6} e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*((b*x+a)^2)^(5/2)*(30*B*a^4*b*d*e^5*x+60*B*a^3*b^2*d^2*e^4*x+120*B*a^2*b^3*d^3*e^3*x+300*B*a*b^4*d^4*e^2
*x+10*A*b^5*d^5*e+10*A*a*b^4*d^4*e^2+5*B*a^4*b*d^2*e^4+10*A*a^3*b^2*d^2*e^4+10*A*a^2*b^3*d^3*e^3+2*B*a^5*d*e^5
+10*B*a^3*b^2*d^3*e^3+20*B*a^2*b^3*d^4*e^2+50*B*a*b^4*d^5*e+150*A*a^2*b^3*d*e^5*x^2+150*A*a*b^4*d^2*e^4*x^2+60
*A*a^4*b*e^6*x+60*A*b^5*d^4*e^2*x+200*A*a^2*b^3*e^6*x^3-360*B*b^5*d*e^5*x^5*ln(e*x+d)-1200*B*b^5*d^3*e^3*x^3*l
n(e*x+d)-360*B*b^5*d^5*e*x*ln(e*x+d)-900*B*b^5*d^2*e^4*x^4*ln(e*x+d)+10*A*a^5*e^6-147*B*b^5*d^6+750*B*a*b^4*d*
e^5*x^4+10*A*a^4*b*d*e^5+400*B*a^2*b^3*d*e^5*x^3+1000*B*a*b^4*d^2*e^4*x^3+60*A*a^3*b^2*d*e^5*x+60*A*a^2*b^3*d^
2*e^4*x+60*A*a*b^4*d^3*e^3*x+150*B*a^3*b^2*d*e^5*x^2+200*A*b^5*d^2*e^4*x^3+200*B*a^3*b^2*e^6*x^3-2200*B*b^5*d^
3*e^3*x^3+150*A*a^3*b^2*e^6*x^2+150*A*b^5*d^3*e^3*x^2+75*B*a^4*b*e^6*x^2-1875*B*b^5*d^4*e^2*x^2+300*B*a*b^4*e^
6*x^5-60*B*ln(e*x+d)*x^6*b^5*e^6+60*A*b^5*e^6*x^5+12*B*a^5*e^6*x-60*B*b^5*d^6*ln(e*x+d)+200*A*a*b^4*d*e^5*x^3-
900*B*b^5*d^4*e^2*x^2*ln(e*x+d)+750*B*a*b^4*d^3*e^3*x^2+300*B*a^2*b^3*d^2*e^4*x^2-822*B*b^5*d^5*e*x-360*B*b^5*
d*e^5*x^5+150*A*a*b^4*e^6*x^4+150*A*b^5*d*e^5*x^4+300*B*a^2*b^3*e^6*x^4-1350*B*b^5*d^2*e^4*x^4)/(b*x+a)^5/e^7/
(e*x+d)^6

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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