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

[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))

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Rubi [A]  time = 0.285352, antiderivative size = 365, normalized size of antiderivative = 1., 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 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]

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 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])

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*}

Mathematica [A]  time = 0.310974, size = 477, normalized size = 1.31 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^3 b^2 e^3 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (A e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )+5 a^4 b e^4 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 a^5 e^5 (5 A e+B (d+6 e x))+10 a b^4 e \left (A e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )+b^5 \left (10 A e \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )-B d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )\right )-60 b^5 B (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \]

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]

-(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 + 1
5*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 + 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)) - 60*b^5*B*(d + e*x)^6*Log[d + e*x]))/(60*e^7*(a
+ b*x)*(d + e*x)^6)

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Maple [B]  time = 0.017, size = 809, normalized size = 2.2 \begin{align*} -{\frac{-360\,B\ln \left ( ex+d \right ) x{b}^{5}{d}^{5}e+10\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+20\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+50\,Ba{b}^{4}{d}^{5}e+10\,Aa{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}-900\,B\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{4}{e}^{2}-1200\,B\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{3}{e}^{3}-900\,B\ln \left ( ex+d \right ){x}^{4}{b}^{5}{d}^{2}{e}^{4}-360\,B\ln \left ( ex+d \right ){x}^{5}{b}^{5}d{e}^{5}+750\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+60\,Ax{a}^{3}{b}^{2}d{e}^{5}+60\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+60\,Axa{b}^{4}{d}^{3}{e}^{3}+30\,Bx{a}^{4}bd{e}^{5}+60\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+300\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+150\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+120\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+300\,Bxa{b}^{4}{d}^{4}{e}^{2}+750\,B{x}^{4}a{b}^{4}d{e}^{5}+200\,A{x}^{3}a{b}^{4}d{e}^{5}+400\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+1000\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+150\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+150\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-60\,B\ln \left ( ex+d \right ){b}^{5}{d}^{6}+60\,A{x}^{5}{b}^{5}{e}^{6}+12\,Bx{a}^{5}{e}^{6}+2\,Bd{e}^{5}{a}^{5}+10\,A{b}^{5}{d}^{5}e+10\,A{a}^{5}{e}^{6}-147\,B{b}^{5}{d}^{6}-60\,B\ln \left ( ex+d \right ){x}^{6}{b}^{5}{e}^{6}-1350\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}-2200\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+150\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+300\,B{x}^{5}a{b}^{4}{e}^{6}-360\,B{x}^{5}{b}^{5}d{e}^{5}+150\,A{x}^{4}a{b}^{4}{e}^{6}+150\,A{x}^{4}{b}^{5}d{e}^{5}+300\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-822\,Bx{b}^{5}{d}^{5}e+200\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+200\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+200\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+60\,Ax{a}^{4}b{e}^{6}+60\,Ax{b}^{5}{d}^{4}{e}^{2}+150\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+75\,B{x}^{2}{a}^{4}b{e}^{6}-1875\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+10\,Ad{e}^{5}{a}^{4}b}{60\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66974, size = 1451, normalized size = 3.98 \begin{align*} \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 )}} \end{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.17514, size = 1176, normalized size = 3.22 \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)*(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