Optimal. Leaf size=457 \[ -\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}+\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {10 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.59, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {732, 814, 843, 715, 112, 110, 117, 116} \[ -\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-b^3 e^3-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {10 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 732
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {10 \int \frac {\left (-\frac {1}{2} b c d (16 c d-15 b e)-\frac {1}{2} c \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{21 c e^3}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} b c d \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+\frac {1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 c^2 e^5}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 c e^6}+\frac {\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{63 c e^6}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{63 c e^6 \sqrt {b x+c x^2}}+\frac {\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{63 c e^6 \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{63 c e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{63 c e^6 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \sqrt {-b} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.28, size = 498, normalized size = 1.09 \[ \frac {2 (x (b+c x))^{5/2} \left (-e \sqrt {x} (b+c x) \left (-b^3 e^3 (d+e x)+3 b^2 c e^2 \left (37 d^2+11 d e x-5 e^2 x^2\right )-b c^2 e \left (240 d^3+64 d^2 e x-31 d e^2 x^2+19 e^3 x^3\right )+c^3 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )+\frac {2 (b+c x) (d+e x) \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{c \sqrt {x}}+i e x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (2 b^4 e^4+13 b^3 c d e^3-159 b^2 c^2 d^2 e^2+272 b c^3 d^3 e-128 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-2 i e x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^4 e^4+7 b^3 c d e^3-135 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )}{63 c e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.12, size = 1170, normalized size = 2.56 \[ \frac {2 \sqrt {\left (c x +b \right ) x}\, \sqrt {e x +d}\, \left (7 c^{6} e^{5} x^{6}+26 b \,c^{5} e^{5} x^{5}-10 c^{6} d \,e^{4} x^{5}+34 b^{2} c^{4} e^{5} x^{4}-41 b \,c^{5} d \,e^{4} x^{4}+16 c^{6} d^{2} e^{3} x^{4}+16 b^{3} c^{3} e^{5} x^{3}-64 b^{2} c^{4} d \,e^{4} x^{3}+80 b \,c^{5} d^{2} e^{3} x^{3}-32 c^{6} d^{3} e^{2} x^{3}+b^{4} c^{2} e^{5} x^{2}-32 b^{3} c^{3} d \,e^{4} x^{2}-47 b^{2} c^{4} d^{2} e^{3} x^{2}+208 b \,c^{5} d^{3} e^{2} x^{2}-128 c^{6} d^{4} e \,x^{2}+2 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{6} e^{5} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+12 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{5} c d \,e^{4} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+\sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{5} c d \,e^{4} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-284 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{4} c^{2} d^{2} e^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+125 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{4} c^{2} d^{2} e^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+b^{4} c^{2} d \,e^{4} x +782 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{3} c^{3} d^{3} e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-510 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{3} c^{3} d^{3} e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-111 b^{3} c^{3} d^{2} e^{3} x -768 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{2} c^{4} d^{4} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+640 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{2} c^{4} d^{4} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+240 b^{2} c^{4} d^{3} e^{2} x +256 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b \,c^{5} d^{5} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-256 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b \,c^{5} d^{5} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-128 b \,c^{5} d^{4} e x \right )}{63 \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{3} e^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________