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

Optimal. Leaf size=368 \[ -\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) \sqrt {d+e x}}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^7 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^7 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^7 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^7 (a+b x)}-\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) \sqrt {d+e x}}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) + (4*b*(b*d - a*e)^5*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^(3/2)) - (30*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(e^7*(a + b*x)*Sqrt[d + e*x]) - (40*b^3*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a +
b*x)) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (12*b^5*(b*d -
a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(7*e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, 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 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)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{7/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{7/2}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{7/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{5/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{3/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 \sqrt {d+e x}}+\frac {15 b^4 (b d-a e)^2 \sqrt {d+e x}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{3/2}}{e^6}+\frac {b^6 (d+e x)^{5/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac {4 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {30 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {40 b^3 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 163, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-42 b^5 (d+e x)^5 (b d-a e)+175 b^4 (d+e x)^4 (b d-a e)^2-700 b^3 (d+e x)^3 (b d-a e)^3-525 b^2 (d+e x)^2 (b d-a e)^4+70 b (d+e x) (b d-a e)^5-7 (b d-a e)^6+5 b^6 (d+e x)^6\right )}{35 e^7 (a+b x) (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(-7*(b*d - a*e)^6 + 70*b*(b*d - a*e)^5*(d + e*x) - 525*b^2*(b*d - a*e)^4*(d + e*x)^2 - 70
0*b^3*(b*d - a*e)^3*(d + e*x)^3 + 175*b^4*(b*d - a*e)^2*(d + e*x)^4 - 42*b^5*(b*d - a*e)*(d + e*x)^5 + 5*b^6*(
d + e*x)^6))/(35*e^7*(a + b*x)*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 31.48, size = 466, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-7 a^6 e^6-70 a^5 b e^5 (d+e x)+42 a^5 b d e^5-105 a^4 b^2 d^2 e^4-525 a^4 b^2 e^4 (d+e x)^2+350 a^4 b^2 d e^4 (d+e x)+140 a^3 b^3 d^3 e^3-700 a^3 b^3 d^2 e^3 (d+e x)+700 a^3 b^3 e^3 (d+e x)^3+2100 a^3 b^3 d e^3 (d+e x)^2-105 a^2 b^4 d^4 e^2+700 a^2 b^4 d^3 e^2 (d+e x)-3150 a^2 b^4 d^2 e^2 (d+e x)^2+175 a^2 b^4 e^2 (d+e x)^4-2100 a^2 b^4 d e^2 (d+e x)^3+42 a b^5 d^5 e-350 a b^5 d^4 e (d+e x)+2100 a b^5 d^3 e (d+e x)^2+2100 a b^5 d^2 e (d+e x)^3+42 a b^5 e (d+e x)^5-350 a b^5 d e (d+e x)^4-7 b^6 d^6+70 b^6 d^5 (d+e x)-525 b^6 d^4 (d+e x)^2-700 b^6 d^3 (d+e x)^3+175 b^6 d^2 (d+e x)^4+5 b^6 (d+e x)^6-42 b^6 d (d+e x)^5\right )}{35 e^6 (d+e x)^{5/2} (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-7*b^6*d^6 + 42*a*b^5*d^5*e - 105*a^2*b^4*d^4*e^2 + 140*a^3*b^3*d^3*e^3 - 105*a^
4*b^2*d^2*e^4 + 42*a^5*b*d*e^5 - 7*a^6*e^6 + 70*b^6*d^5*(d + e*x) - 350*a*b^5*d^4*e*(d + e*x) + 700*a^2*b^4*d^
3*e^2*(d + e*x) - 700*a^3*b^3*d^2*e^3*(d + e*x) + 350*a^4*b^2*d*e^4*(d + e*x) - 70*a^5*b*e^5*(d + e*x) - 525*b
^6*d^4*(d + e*x)^2 + 2100*a*b^5*d^3*e*(d + e*x)^2 - 3150*a^2*b^4*d^2*e^2*(d + e*x)^2 + 2100*a^3*b^3*d*e^3*(d +
e*x)^2 - 525*a^4*b^2*e^4*(d + e*x)^2 - 700*b^6*d^3*(d + e*x)^3 + 2100*a*b^5*d^2*e*(d + e*x)^3 - 2100*a^2*b^4*
d*e^2*(d + e*x)^3 + 700*a^3*b^3*e^3*(d + e*x)^3 + 175*b^6*d^2*(d + e*x)^4 - 350*a*b^5*d*e*(d + e*x)^4 + 175*a^
2*b^4*e^2*(d + e*x)^4 - 42*b^6*d*(d + e*x)^5 + 42*a*b^5*e*(d + e*x)^5 + 5*b^6*(d + e*x)^6))/(35*e^6*(d + e*x)^
(5/2)*(a*e + b*e*x))

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fricas [A]  time = 0.43, size = 388, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (5 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 3584 \, a b^{5} d^{5} e - 4480 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} - 280 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 6 \, {\left (2 \, b^{6} d e^{5} - 7 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (8 \, b^{6} d^{2} e^{4} - 28 \, a b^{5} d e^{5} + 35 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (16 \, b^{6} d^{3} e^{3} - 56 \, a b^{5} d^{2} e^{4} + 70 \, a^{2} b^{4} d e^{5} - 35 \, a^{3} b^{3} e^{6}\right )} x^{3} - 15 \, {\left (128 \, b^{6} d^{4} e^{2} - 448 \, a b^{5} d^{3} e^{3} + 560 \, a^{2} b^{4} d^{2} e^{4} - 280 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} - 10 \, {\left (256 \, b^{6} d^{5} e - 896 \, a b^{5} d^{4} e^{2} + 1120 \, a^{2} b^{4} d^{3} e^{3} - 560 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 7 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}

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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*b^6*e^6*x^6 - 1024*b^6*d^6 + 3584*a*b^5*d^5*e - 4480*a^2*b^4*d^4*e^2 + 2240*a^3*b^3*d^3*e^3 - 280*a^4*
b^2*d^2*e^4 - 28*a^5*b*d*e^5 - 7*a^6*e^6 - 6*(2*b^6*d*e^5 - 7*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 28*a*b^5*d*e
^5 + 35*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 56*a*b^5*d^2*e^4 + 70*a^2*b^4*d*e^5 - 35*a^3*b^3*e^6)*x^3 - 15
*(128*b^6*d^4*e^2 - 448*a*b^5*d^3*e^3 + 560*a^2*b^4*d^2*e^4 - 280*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 - 10*(25
6*b^6*d^5*e - 896*a*b^5*d^4*e^2 + 1120*a^2*b^4*d^3*e^3 - 560*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 7*a^5*b*e^6)
*x)*sqrt(e*x + d)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [B]  time = 0.30, size = 626, normalized size = 1.70 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} e^{42} \mathrm {sgn}\left (b x + a\right ) - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d e^{42} \mathrm {sgn}\left (b x + a\right ) + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{2} e^{42} \mathrm {sgn}\left (b x + a\right ) - 700 \, \sqrt {x e + d} b^{6} d^{3} e^{42} \mathrm {sgn}\left (b x + a\right ) + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} e^{43} \mathrm {sgn}\left (b x + a\right ) - 350 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d e^{43} \mathrm {sgn}\left (b x + a\right ) + 2100 \, \sqrt {x e + d} a b^{5} d^{2} e^{43} \mathrm {sgn}\left (b x + a\right ) + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{44} \mathrm {sgn}\left (b x + a\right ) - 2100 \, \sqrt {x e + d} a^{2} b^{4} d e^{44} \mathrm {sgn}\left (b x + a\right ) + 700 \, \sqrt {x e + d} a^{3} b^{3} e^{45} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 10 \, {\left (x e + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 300 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 50 \, {\left (x e + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 450 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 100 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 300 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 100 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 75 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 50 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, {\left (x e + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{5 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^6*e^42*sgn(b*x + a) - 42*(x*e + d)^(5/2)*b^6*d*e^42*sgn(b*x + a) + 175*(x*e + d)^(3/
2)*b^6*d^2*e^42*sgn(b*x + a) - 700*sqrt(x*e + d)*b^6*d^3*e^42*sgn(b*x + a) + 42*(x*e + d)^(5/2)*a*b^5*e^43*sgn
(b*x + a) - 350*(x*e + d)^(3/2)*a*b^5*d*e^43*sgn(b*x + a) + 2100*sqrt(x*e + d)*a*b^5*d^2*e^43*sgn(b*x + a) + 1
75*(x*e + d)^(3/2)*a^2*b^4*e^44*sgn(b*x + a) - 2100*sqrt(x*e + d)*a^2*b^4*d*e^44*sgn(b*x + a) + 700*sqrt(x*e +
d)*a^3*b^3*e^45*sgn(b*x + a))*e^(-49) - 2/5*(75*(x*e + d)^2*b^6*d^4*sgn(b*x + a) - 10*(x*e + d)*b^6*d^5*sgn(b
*x + a) + b^6*d^6*sgn(b*x + a) - 300*(x*e + d)^2*a*b^5*d^3*e*sgn(b*x + a) + 50*(x*e + d)*a*b^5*d^4*e*sgn(b*x +
a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 450*(x*e + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) - 100*(x*e + d)*a^2*b^4*d^3*e^
2*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 300*(x*e + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) + 100*(x*e + d)*
a^3*b^3*d^2*e^3*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 75*(x*e + d)^2*a^4*b^2*e^4*sgn(b*x + a) - 50*
(x*e + d)*a^4*b^2*d*e^4*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) + 10*(x*e + d)*a^5*b*e^5*sgn(b*x + a) -
6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(5/2)

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maple [A]  time = 0.05, size = 393, normalized size = 1.07 \begin {gather*} -\frac {2 \left (-5 b^{6} e^{6} x^{6}-42 a \,b^{5} e^{6} x^{5}+12 b^{6} d \,e^{5} x^{5}-175 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}-40 b^{6} d^{2} e^{4} x^{4}-700 a^{3} b^{3} e^{6} x^{3}+1400 a^{2} b^{4} d \,e^{5} x^{3}-1120 a \,b^{5} d^{2} e^{4} x^{3}+320 b^{6} d^{3} e^{3} x^{3}+525 a^{4} b^{2} e^{6} x^{2}-4200 a^{3} b^{3} d \,e^{5} x^{2}+8400 a^{2} b^{4} d^{2} e^{4} x^{2}-6720 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+70 a^{5} b \,e^{6} x +700 a^{4} b^{2} d \,e^{5} x -5600 a^{3} b^{3} d^{2} e^{4} x +11200 a^{2} b^{4} d^{3} e^{3} x -8960 a \,b^{5} d^{4} e^{2} x +2560 b^{6} d^{5} e x +7 a^{6} e^{6}+28 a^{5} b d \,e^{5}+280 a^{4} b^{2} d^{2} e^{4}-2240 a^{3} b^{3} d^{3} e^{3}+4480 a^{2} b^{4} d^{4} e^{2}-3584 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{35 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{5} e^{7}} \end {gather*}

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)^(7/2),x)

[Out]

-2/35/(e*x+d)^(5/2)*(-5*b^6*e^6*x^6-42*a*b^5*e^6*x^5+12*b^6*d*e^5*x^5-175*a^2*b^4*e^6*x^4+140*a*b^5*d*e^5*x^4-
40*b^6*d^2*e^4*x^4-700*a^3*b^3*e^6*x^3+1400*a^2*b^4*d*e^5*x^3-1120*a*b^5*d^2*e^4*x^3+320*b^6*d^3*e^3*x^3+525*a
^4*b^2*e^6*x^2-4200*a^3*b^3*d*e^5*x^2+8400*a^2*b^4*d^2*e^4*x^2-6720*a*b^5*d^3*e^3*x^2+1920*b^6*d^4*e^2*x^2+70*
a^5*b*e^6*x+700*a^4*b^2*d*e^5*x-5600*a^3*b^3*d^2*e^4*x+11200*a^2*b^4*d^3*e^3*x-8960*a*b^5*d^4*e^2*x+2560*b^6*d
^5*e*x+7*a^6*e^6+28*a^5*b*d*e^5+280*a^4*b^2*d^2*e^4-2240*a^3*b^3*d^3*e^3+4480*a^2*b^4*d^4*e^2-3584*a*b^5*d^5*e
+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [B]  time = 0.86, size = 647, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} a}{15 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (15 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 8960 \, a b^{4} d^{5} e - 8960 \, a^{2} b^{3} d^{4} e^{2} + 3360 \, a^{3} b^{2} d^{3} e^{3} - 280 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} - 3 \, {\left (12 \, b^{5} d e^{5} - 35 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 35 \, a b^{4} d e^{5} + 35 \, a^{2} b^{3} e^{6}\right )} x^{4} - 10 \, {\left (96 \, b^{5} d^{3} e^{3} - 280 \, a b^{4} d^{2} e^{4} + 280 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} - 15 \, {\left (384 \, b^{5} d^{4} e^{2} - 1120 \, a b^{4} d^{3} e^{3} + 1120 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} - 5 \, {\left (1536 \, b^{5} d^{5} e - 4480 \, a b^{4} d^{4} e^{2} + 4480 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 7 \, a^{5} e^{6}\right )} x\right )} b}{105 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt {e x + d}} \end {gather*}

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)^(7/2),x, algorithm="maxima")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 +
30*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e - 320*a*b^4*
d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*a/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x
+ d)) + 2/105*(15*b^5*e^6*x^6 - 3072*b^5*d^6 + 8960*a*b^4*d^5*e - 8960*a^2*b^3*d^4*e^2 + 3360*a^3*b^2*d^3*e^3
- 280*a^4*b*d^2*e^4 - 14*a^5*d*e^5 - 3*(12*b^5*d*e^5 - 35*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 35*a*b^4*d*e^
5 + 35*a^2*b^3*e^6)*x^4 - 10*(96*b^5*d^3*e^3 - 280*a*b^4*d^2*e^4 + 280*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 -
15*(384*b^5*d^4*e^2 - 1120*a*b^4*d^3*e^3 + 1120*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 - 5*(1
536*b^5*d^5*e - 4480*a*b^4*d^4*e^2 + 4480*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4 + 140*a^4*b*d*e^5 + 7*a^5*e^6
)*x)*b/((e^9*x^2 + 2*d*e^8*x + d^2*e^7)*sqrt(e*x + d))

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mupad [B]  time = 3.18, size = 455, normalized size = 1.24 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^5\,x^6}{7\,e^3}-\frac {\frac {2\,a^6\,e^6}{5}+\frac {8\,a^5\,b\,d\,e^5}{5}+16\,a^4\,b^2\,d^2\,e^4-128\,a^3\,b^3\,d^3\,e^3+256\,a^2\,b^4\,d^4\,e^2-\frac {1024\,a\,b^5\,d^5\,e}{5}+\frac {2048\,b^6\,d^6}{35}}{b\,e^9}-\frac {x\,\left (140\,a^5\,b\,e^6+1400\,a^4\,b^2\,d\,e^5-11200\,a^3\,b^3\,d^2\,e^4+22400\,a^2\,b^4\,d^3\,e^3-17920\,a\,b^5\,d^4\,e^2+5120\,b^6\,d^5\,e\right )}{35\,b\,e^9}+\frac {b^2\,x^3\,\left (40\,a^3\,e^3-80\,a^2\,b\,d\,e^2+64\,a\,b^2\,d^2\,e-\frac {128\,b^3\,d^3}{7}\right )}{e^6}+\frac {b^4\,x^5\,\left (\frac {12\,a\,e}{5}-\frac {24\,b\,d}{35}\right )}{e^4}+\frac {b^3\,x^4\,\left (10\,a^2\,e^2-8\,a\,b\,d\,e+\frac {16\,b^2\,d^2}{7}\right )}{e^5}-\frac {x^2\,\left (30\,a^4\,b^2\,e^6-240\,a^3\,b^3\,d\,e^5+480\,a^2\,b^4\,d^2\,e^4-384\,a\,b^5\,d^3\,e^3+\frac {768\,b^6\,d^4\,e^2}{7}\right )}{b\,e^9}\right )}{x^3\,\sqrt {d+e\,x}+\frac {a\,d^2\,\sqrt {d+e\,x}}{b\,e^2}+\frac {x^2\,\left (a\,e^9+2\,b\,d\,e^8\right )\,\sqrt {d+e\,x}}{b\,e^9}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}} \end {gather*}

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)^(7/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^5*x^6)/(7*e^3) - ((2*a^6*e^6)/5 + (2048*b^6*d^6)/35 + 256*a^2*b^4*d^4*e
^2 - 128*a^3*b^3*d^3*e^3 + 16*a^4*b^2*d^2*e^4 - (1024*a*b^5*d^5*e)/5 + (8*a^5*b*d*e^5)/5)/(b*e^9) - (x*(140*a^
5*b*e^6 + 5120*b^6*d^5*e - 17920*a*b^5*d^4*e^2 + 1400*a^4*b^2*d*e^5 + 22400*a^2*b^4*d^3*e^3 - 11200*a^3*b^3*d^
2*e^4))/(35*b*e^9) + (b^2*x^3*(40*a^3*e^3 - (128*b^3*d^3)/7 + 64*a*b^2*d^2*e - 80*a^2*b*d*e^2))/e^6 + (b^4*x^5
*((12*a*e)/5 - (24*b*d)/35))/e^4 + (b^3*x^4*(10*a^2*e^2 + (16*b^2*d^2)/7 - 8*a*b*d*e))/e^5 - (x^2*(30*a^4*b^2*
e^6 + (768*b^6*d^4*e^2)/7 - 384*a*b^5*d^3*e^3 - 240*a^3*b^3*d*e^5 + 480*a^2*b^4*d^2*e^4))/(b*e^9)))/(x^3*(d +
e*x)^(1/2) + (a*d^2*(d + e*x)^(1/2))/(b*e^2) + (x^2*(a*e^9 + 2*b*d*e^8)*(d + e*x)^(1/2))/(b*e^9) + (d*x*(2*a*e
+ b*d)*(d + e*x)^(1/2))/(b*e^2))

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sympy [F(-1)]  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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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